  ok   R4: nu_3(V(y^2, yz, z^3)) = <X^3, X^2Y, X^2Z, XZ^2>
=== R4: B = V(y^2, yz, z^3) (local ideal (u^2, uw, w^3) at [X]); seven normal forms N1..N7 of (h, B5)
--- R4 N6
  ok   R4 N6: the graded normal form is closed with H = [1, 3, 6, 4, 3, 2, 1, 1]
  ok   R4 N6: one lift per pivot of D_4..D_7
    128 closedness conditions in 82 tail coefficients; 55 eliminated with constant pivots; residual equations:
       (1/14) * (-2*P_510 + 14*G_220 + 7*G_112) * P_203
       (1/14) * P_203 * (2*P_510 + 7*G_112 + 84*G_004)
  ok   R4 N6: on {P_203 != 0} the residual equations are triangular in ['G_220', 'G_004'] with diagonal (const)*P_203: a graph (irreducible)
  ok   R4 N6: on {P_203 = 0}: no equations remain
  ok   R4 N6, component W1 (P_203 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N6, component W1 (P_203 != 0) anchor (A): fixed {'P_203': 1}, solved ['G_220', 'G_004']; member closed with H and the prescribed graded data: True; a triple of I of local length 39 (ds), stabilization (39, 11)
      anchor generators: {'P': 'X^7 - 2*X^3*Y*Z + X^2*Z^3 + 2*X*Y^2 - 2*Y^3 - 2*X*Y*Z - 2*Y^2*Z + Y*Z^2', 'G': 'X^4*Y - 2*X^3*Z^2 - 2*X*Y^2 - Y^3 + 2*Y^2*Z + 2*Z^3'}
      anchor triple: ['x^8 + x^7*y - x^6*y^2 - x^5*y^3 + 2*x^4*y^4 - 3*x^3*y^5 + 2*x^2*y^6 + x*y^7 - 3*y^8 + x^7*z + 3*x^6*y*z - 3*x^5*y^2*z + 2*x^4*y^3*z - 2*x^3*y^4*z + 3*x^2*y^5*z + 3*x*y^6*z + 2*y^7*z + 2*x^6*z^2 - 2*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + x^2*y^4*z^2 - 3*x*y^5*z^2 + 3*y^6*z^2 + x^4*y*z^3 - 2*x^3*y^2*z^3 - y^5*z^3 + 3*x^3*y*z^4 - 2*x^2*y^2*z^4 + 3*x*y^3*z^4 + x^3*z^5 + 3*x^2*y*z^5 - 2*x*y^2*z^5 + 2*y^3*z^5 - 2*x^2*z^6 + x*y*z^6 + y^2*z^6 - 2*x*z^7 - y*z^7 - 1/180*x^7 + 3*x^6*y - x^5*y^2 + 2*x^2*y^5 + 3*x*y^6 - y^7 + 2*x^5*y*z + 3*x^4*y^2*z - 3*x^3*y^3*z - 3*x^2*y^4*z - x*y^5*z + 3*y^6*z - x^5*z^2 + x^4*y*z^2 + 2*x^3*y^2*z^2 - x^2*y^3*z^2 + x*y^4*z^2 - 2*x^4*z^3 + 3*x^3*y*z^3 - 3*x^2*y^2*z^3 + 3*x*y^3*z^3 - 3*x*y^2*z^4 - 3*x*y*z^5 - 3*y^2*z^5 - y*z^6 - 1/140*x^6 - 3*x^5*y - x^4*y^2 - 2*x^3*y^3 - x^2*y^4 - 3*x*y^5 + 2*x^5*z - x^4*y*z - 2*x^3*y^2*z + x*y^4*z + 3*y^5*z + x^4*z^2 + x^2*y^2*z^2 - 3*y^4*z^2 - 3*x^3*z^3 - 2*x^2*y*z^3 - 2*x*y^2*z^3 - y^3*z^3 + 2*x^2*z^4 - 2*x*y*z^4 - y^2*z^4 - 3*x*z^5 - 3*y*z^5 + 2*z^6 + 1/140*x^5 + 23/6*x^4*y - 2*x^3*y^2 + 3*x^2*y^3 - 2*x*y^4 + 2*y^5 + x^4*z + 2*x^3*y*z + 3*x^2*y^2*z - 3*x*y^3*z + y^4*z + 3*x^3*z^2 + 3*x*y^2*z^2 - 2*y^3*z^2 + 3*x^2*z^3 + 2*x*y*z^3 - y^2*z^3 - 3*x*z^4 + y*z^4 + 3*z^5 - 2*x^3*y - 2*x^2*y^2 + 2*x*y^3 + 3*y^4 - 3*x^2*y*z - 3*x*y^2*z - 2*x^2*z^2 - x*y*z^2 + 3*y^2*z^2 + 2*y*z^3 - 2*z^4 - x*y^2 - 2*y^3 - 2*y*z^2 - 3*z^3', '3*x^8 - 3*x^6*y^2 - 3*x^5*y^3 - 3*x^2*y^6 + 2*y^8 - 2*x^7*z + x^6*y*z - x^5*y^2*z + 3*x^3*y^4*z + 3*x*y^6*z + y^7*z + 2*x^6*z^2 + 3*x^5*y*z^2 + 2*x^4*y^2*z^2 - 3*x^3*y^3*z^2 + 2*x^2*y^4*z^2 + y^6*z^2 + x^5*z^3 - 3*x^4*y*z^3 + x^3*y^2*z^3 + x^2*y^3*z^3 - 2*x*y^4*z^3 - y^5*z^3 - 3*x^4*z^4 + 3*x^3*y*z^4 - 2*x^2*y^2*z^4 - 2*x*y^3*z^4 - 3*y^4*z^4 - x^3*z^5 - x^2*y*z^5 + 2*x*y^2*z^5 + 2*y^3*z^5 - 2*x^2*z^6 + 2*x*y*z^6 + y^2*z^6 - x*z^7 + 2*y*z^7 + 2*z^8 + 17/840*x^7 - 3*x^6*y - x^5*y^2 + x^4*y^3 - 3*x^3*y^4 - 2*x^2*y^5 + 3*y^7 - x^6*z - x^5*y*z - 2*x^4*y^2*z - 3*x^3*y^3*z + 3*x^2*y^4*z - 2*x*y^5*z - 2*y^6*z - 2*x^5*z^2 + 3*x^4*y*z^2 + x^3*y^2*z^2 + 3*x^2*y^3*z^2 + 3*y^5*z^2 - x^4*z^3 - 3*x^3*y*z^3 - 3*x^2*y^2*z^3 - 2*x*y^3*z^3 + 3*y^4*z^3 + x^3*z^4 + 3*x^2*y*z^4 - 2*x*y^2*z^4 + 2*y^3*z^4 - 2*x*y*z^5 + 2*y^2*z^5 - 2*x*z^6 + y*z^6 + 2*z^7 - 1/210*x^6 + 2*x^4*y^2 + x^3*y^3 - 3*x*y^5 + 2*y^6 + x^4*y*z + 3*x^3*y^2*z + 3*x^2*y^3*z + x*y^4*z + y^5*z + 2*x^4*z^2 + 2*x^3*y*z^2 - 3*x*y^3*z^2 + y^4*z^2 - 3*x^3*z^3 - 2*x^2*y*z^3 - 2*x*y^2*z^3 + 2*y^3*z^3 - x^2*z^4 - 3*y^2*z^4 + x*z^5 + 2*y*z^5 + 2*z^6 - 1/140*x^5 + 3/4*x^4*y - 2*x^3*y^2 - x^2*y^3 - x*y^4 - 2*y^5 + 2*x^4*z + 2*x^3*y*z + 3*x^2*y^2*z - 2*x*y^3*z + 3*y^4*z + x^3*z^2 - 3*x^2*y*z^2 + 3*x*y^2*z^2 - 2*y^3*z^2 - 3*x^2*z^3 - x*y*z^3 - 2*y*z^4 + z^5 - 2*x^3*y - x^2*y^2 + 2*x*y^3 - 3*y^4 + x*y^2*z - 2*x^2*z^2 - 3*x*y*z^2 + 3*y^2*z^2 + 2*x*z^3 - 3*y*z^3 + 3*z^4 - 3*x*y^2 + 3*y^3 - 2*x*y*z + y*z^2 + z^3', 'x^8 - 3*x^7*y - 2*x^6*y^2 - x^5*y^3 + x^4*y^4 + 3*x^3*y^5 + 2*x^2*y^6 + 3*x*y^7 + 2*y^8 - x^4*y^3*z + 3*x^3*y^4*z - 3*x*y^6*z - 2*y^7*z - x^5*y*z^2 - 3*x^4*y^2*z^2 + 2*x^3*y^3*z^2 - 3*x^2*y^4*z^2 + 2*x*y^5*z^2 - 2*y^6*z^2 + 3*x^5*z^3 + x^4*y*z^3 - 2*x^3*y^2*z^3 - 3*x^2*y^3*z^3 + 3*x*y^4*z^3 - 2*y^5*z^3 - 2*x^4*z^4 - x^3*y*z^4 + 2*x^2*y^2*z^4 - 3*x*y^3*z^4 + 2*y^4*z^4 - 2*x^3*z^5 - 3*x^2*y*z^5 + 3*x*y^2*z^5 + 2*x^2*z^6 + x*y*z^6 + 3*y^2*z^6 + x*z^7 - 11/840*x^7 - 2*x^6*y - 3*x^5*y^2 + 2*x^4*y^3 + 2*x^3*y^4 + x^2*y^5 - 2*x*y^6 + 3*y^7 + x^6*z + 3*x^5*y*z + 3*x^4*y^2*z - 2*x^3*y^3*z + 3*x^2*y^4*z - x*y^5*z - 3*y^6*z + x^5*z^2 - 2*x^3*y^2*z^2 - x*y^4*z^2 + 2*y^5*z^2 - 3*x^4*z^3 - 2*x*y^3*z^3 - y^4*z^3 + 2*x^3*z^4 + 3*x^2*y*z^4 - 3*x*y^2*z^4 - y^3*z^4 - x^2*z^5 - 2*x*y*z^5 + 2*y^2*z^5 - 3*x*z^6 - y*z^6 + 2*z^7 + 1/420*x^6 - 2*x^5*y + 3*x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - 2*x*y^5 - y^6 + 3*x^5*z + 2*x^4*y*z + x^3*y^2*z + 3*x^2*y^3*z + 2*y^5*z + x^4*z^2 + 2*x^2*y^2*z^2 + x*y^3*z^2 + 2*y^4*z^2 - x^3*z^3 + x^2*y*z^3 + 2*x*y^2*z^3 - 2*y^3*z^3 - x^2*z^4 + 2*x*y*z^4 + 2*x*z^5 - 3*y*z^5 + 1/210*x^5 + 31/12*x^4*y + x^3*y^2 + x^2*y^3 + 3*x*y^4 - 2*y^5 + 2*x^4*z - x^3*y*z + x*y^3*z - 3*y^4*z + 3*x^3*z^2 - x^2*y*z^2 + x*y^2*z^2 + 2*y^3*z^2 + 2*x^2*z^3 + 2*y^2*z^3 + 2*y*z^4 + 2*z^5 + 2*x^3*y - 2*y^4 + 3*x^2*y*z - x*y^2*z + 3*y^3*z + 2*x^2*z^2 + x*y*z^2 - y^2*z^2 + 2*x*z^3 + 2*y*z^3 - 2*x*y^2 - 3*y^3 + x*y*z - y^2*z - z^3']
  ok   R4 N6, component W2 (P_203 = 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.2 s
  ok   R4 N6, component W2 (P_203 = 0) anchor (A): fixed {'P_203': 0}, solved []; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 12)
      anchor generators: {'P': 'X^7 - 2*X^5*Y + X^5*Z + 20/21*X^3*Y^2 - X^4*Z - 20/21*X^3*Y*Z - 2*X^3*Z^2 - 2*X^2*Y^2 - 40/441*X*Y^3 + 20/147*X*Y^2*Z + X^2*Z^2 - 113/49*X*Y*Z^2 - X*Z^3 - 2*X*Y^2 + Y^3 + X*Y*Z + Y*Z^2', 'G': 'X^4*Y - 2*X^3*Z^2 + 2*X^2*Y^2 + 2*X^2*Y*Z + 2*X^2*Z^2 - 2*X*Y*Z^2 - 2*X*Y^2 - 2*Y^3 + X*Y*Z + 2*Y*Z^2'}
      anchor triple: ['x^8 + 2*x^7*y - x^6*y^2 + 2*x^5*y^3 - x^4*y^4 - 3*x^3*y^5 + x^2*y^6 + x*y^7 + 3*y^8 + 2*x^7*z - 3*x^6*y*z - 2*x^5*y^2*z - 3*x^4*y^3*z + 2*x^3*y^4*z + x^2*y^5*z + x*y^6*z + 3*y^7*z + 3*x^6*z^2 + 3*x^5*y*z^2 - 3*x^4*y^2*z^2 + 3*x^3*y^3*z^2 + 2*x*y^5*z^2 - 2*y^6*z^2 - x^5*z^3 - x^3*y^2*z^3 + 3*x*y^4*z^3 + 3*y^5*z^3 + x^4*z^4 - 2*x^3*y*z^4 + x^2*y^2*z^4 + x*y^3*z^4 + y^4*z^4 + x^3*z^5 + 2*x^2*y*z^5 + 2*x*y^2*z^5 - x*y*z^6 - 2*y^2*z^6 + x*z^7 + z^8 + 6949/185220*x^7 + x^4*y^3 + x^3*y^4 + x^2*y^5 - 2*x*y^6 - 2*y^7 + x^6*z + 3*x^5*y*z - 3*x^4*y^2*z + 2*x^3*y^3*z + x^2*y^4*z + 2*x*y^5*z + 2*y^6*z + 3*x^5*z^2 - 3*x^3*y^2*z^2 + 2*x^2*y^3*z^2 + 2*x*y^4*z^2 + x^4*z^3 - x^3*y*z^3 - x^2*y^2*z^3 + 3*x*y^3*z^3 - y^4*z^3 + 3*x^2*y*z^4 + 2*x*y^2*z^4 + 3*y^3*z^4 - 3*x^2*z^5 + x*y*z^5 + y^2*z^5 + 3*x*z^6 + 2*z^7 + 2873/37044*x^6 + 2*x^5*y - x^3*y^3 - x*y^5 + y^6 + 2*x^5*z + x^3*y^2*z - x^2*y^3*z + 2*x*y^4*z - 2*x^4*z^2 - 3*x^3*y*z^2 + 3*y^4*z^2 - x^3*z^3 - 2*x^2*y*z^3 + x*y^2*z^3 - y^3*z^3 - 2*x^2*z^4 + x*y*z^4 + 2*y^2*z^4 - 3*x*z^5 + 3*y*z^5 - z^6 - 34/441*x^5 + 9/4*x^4*y + 3*x^3*y^2 + x^2*y^3 + x*y^4 + y^5 + x^4*z + 3*x^3*y*z - 2*x^2*y^2*z - 3*x*y^3*z - 2*y^4*z + 2*x^2*y*z^2 - x*y^2*z^2 - y^3*z^2 - y^2*z^3 + x*z^4 - 2*y*z^4 + z^5 - 11/6*x^3*y - 3*x^2*y^2 - x*y^3 - 1/3*x^3*z - 3*x^2*y*z - y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - 2*y^2*z^2 - 2*x*z^3 - 2*y*z^3 - 3*z^4 + 2*y^3 + 2*x*y*z - 3*y^2*z + 3*y*z^2 - z^3', '2*x^7*y - 2*x^6*y^2 + x^5*y^3 + 3*x^4*y^4 - 2*x^3*y^5 - x^2*y^6 - 3*x*y^7 - 2*x^7*z - 2*x^6*y*z - 2*x^5*y^2*z + 2*x^4*y^3*z + x^3*y^4*z - x^2*y^5*z - 2*x*y^6*z - y^7*z + 2*x^6*z^2 + 2*x^5*y*z^2 - x^4*y^2*z^2 + x^3*y^3*z^2 - 2*x^2*y^4*z^2 - y^6*z^2 - 3*x^5*z^3 - 2*x^4*y*z^3 - 3*x^3*y^2*z^3 - 3*x^2*y^3*z^3 - 2*x*y^4*z^3 - 2*y^5*z^3 + 2*x^4*z^4 + x^3*y*z^4 + x^2*y^2*z^4 + x*y^3*z^4 - y^4*z^4 - x^3*z^5 + 2*x^2*y*z^5 - 3*y^3*z^5 - 3*x^2*z^6 - 3*x*y*z^6 + 2*y^2*z^6 - 3*x*z^7 + 3*y*z^7 - 2*z^8 - 1391/123480*x^7 - 3*x^6*y - 2*x^5*y^2 - 2*x^4*y^3 + x^3*y^4 - 2*x^2*y^5 - 3*x*y^6 + y^7 + 3*x^6*z - 2*x^5*y*z - x^4*y^2*z - 2*x^2*y^4*z - 2*x*y^5*z - 3*y^6*z - x^4*y*z^2 - x^3*y^2*z^2 - 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 + 2*x^4*z^3 - 2*x^3*y*z^3 + 2*x^2*y^2*z^3 - 3*x*y^3*z^3 - y^4*z^3 - x^3*z^4 + 2*x*y^2*z^4 - y^3*z^4 + x^2*z^5 - x*y*z^5 + y^2*z^5 + 3*x*z^6 - 2*y*z^6 + 3*z^7 - 107/1323*x^6 - x^4*y^2 - 2*x^3*y^3 + 3*x^2*y^4 - 2*x*y^5 + 3*y^6 + 2*x^4*y*z + x^3*y^2*z - x^2*y^3*z + x*y^4*z + x^4*z^2 - 3*x^3*y*z^2 + 3*x*y^3*z^2 - y^4*z^2 + 2*x^3*z^3 + 3*x*y^2*z^3 + y^2*z^4 - 3*x*z^5 + 3*y*z^5 - 2*z^6 - 13/147*x^5 - 29/12*x^4*y - 2*x^3*y^2 + x^2*y^3 - 3*x*y^4 - 2*y^5 - x^4*z + 3*x^3*y*z + y^4*z - 3*x^3*z^2 - 3*x^2*y*z^2 + 2*x*y^2*z^2 - 3*x^2*z^3 + x*y*z^3 - 3*y^2*z^3 + 3*x*z^4 + 2*y*z^4 + 3*z^5 - 3/2*x^3*y - 2*x^2*y^2 + 3*x*y^3 + y^4 + 1/3*x^3*z - 3*x*y^2*z + 3*y^3*z - x^2*z^2 - 2*x*y*z^2 - y^2*z^2 - 3*z^4 + 3*x*y^2 - y^3 - 2*x*y*z + 2*y^2*z + y*z^2 + 3*z^3', 'x^8 + 2*x^7*y + 2*x^6*y^2 - x^5*y^3 + 2*x^4*y^4 - 3*x^3*y^5 - 2*x^2*y^6 - 3*x*y^7 - 3*y^8 + x^7*z + 2*x^6*y*z - 2*x^4*y^3*z - 3*x^3*y^4*z + 3*x^2*y^5*z - 2*x*y^6*z + 2*y^7*z - x^6*z^2 - 2*x^5*y*z^2 - x^4*y^2*z^2 + 2*x^3*y^3*z^2 + x^2*y^4*z^2 + 3*x*y^5*z^2 + 2*x^5*z^3 - 2*x^4*y*z^3 - 3*x^3*y^2*z^3 + 3*x*y^4*z^3 + y^5*z^3 - 2*x^4*z^4 + x^2*y^2*z^4 - 3*x*y^3*z^4 + y^4*z^4 + x^3*z^5 + x^2*y*z^5 + 2*x*y^2*z^5 + 2*x^2*z^6 - x*y*z^6 - 2*y*z^7 - 3*z^8 + 2837/16464*x^7 + x^6*y - 3*x^5*y^2 - 2*x^4*y^3 - x^3*y^4 + x^2*y^5 - x*y^6 - 2*y^7 - x^5*y*z - 2*x^4*y^2*z - 3*x^3*y^3*z - 2*x^2*y^4*z + 3*x*y^5*z + y^6*z - x^5*z^2 + x^4*y*z^2 - 2*x^3*y^2*z^2 - x^2*y^3*z^2 - 3*x*y^4*z^2 + 3*y^5*z^2 + 3*x^4*z^3 + 2*x^3*y*z^3 - 2*x^2*y^2*z^3 + 3*x*y^3*z^3 + 2*y^4*z^3 + 3*x^3*z^4 - x^2*y*z^4 - x*y^2*z^4 + 2*y^3*z^4 - 3*x^2*z^5 - x*y*z^5 - 3*y^2*z^5 - x*z^6 - 3*y*z^6 + 3*z^7 - 5339/61740*x^6 + 2*x^5*y - 3*x^4*y^2 + 2*x^3*y^3 + 2*x^2*y^4 + 3*y^6 - 3*x^5*z - x^4*y*z - 3*x^3*y^2*z + x*y^4*z - 3*x^3*y*z^2 + 2*x^2*y^2*z^2 + 2*x*y^3*z^2 + x^3*z^3 + 2*x^2*y*z^3 + 3*x*y^2*z^3 - y^3*z^3 + x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - 2*x*z^5 + 107/588*x^5 - 25/24*x^4*y + x^3*y^2 - 2*x^2*y^3 + 2*x*y^4 - y^5 + 2*x^4*z - 3*x^3*y*z - 3*x*y^3*z + 3*y^4*z + x^3*z^2 - 3*x^2*y*z^2 - 3*y^3*z^2 + x^2*z^3 + 2*x*y*z^3 + 2*y^2*z^3 - 3*x*z^4 - 3*z^5 + 4*x^3*y - x^2*y^2 + 1/2*x^3*z + x^2*y*z - 3*x*y^2*z - 2*y^3*z + 2*x^2*z^2 + 2*x*y*z^2 + 2*y^2*z^2 - x*z^3 - 2*y*z^3 + 2*z^4 - 3*x*y^2 - 2*y^3 - 3*x*y*z - y^2*z + 3*y*z^2 - 2*z^3']
--- R4 N2
  ok   R4 N2: the graded normal form is closed with H = [1, 3, 6, 4, 3, 2, 1, 1]
  ok   R4 N2: one lift per pivot of D_4..D_7
    128 closedness conditions in 82 tail coefficients; 54 eliminated with constant pivots; residual equations:
       (-3) * G_220 * P_303
       (3/7) * P_303 * (P_510 - 7*G_112)
       (-1/14) * (-3*P_510*P_303*G_310 - 7*P_303*G_211^2 - 21*P_303*G_310*G_112 + 28*P_203*G_220 + 42*P_303*G_120)
       (-3) * G_030 * P_303
       (-1/14) * P_303 * (-P_510*G_211 - 7*G_211*G_112 + 42*G_021)
       (-1/70) * (5*P_501*P_303*G_211 - 105*P_303*G_211*G_103 - 1260*P_303*G_310*G_004 - 6*P_303*P_410 - 10*P_510*P_203 + 70*P_203*G_112 + 210*P_303*G_012)
       (-6/7) * G_220 * P_303
       (3/49) * P_303 * (P_510 - 7*G_112)
  ok   R4 N2: on {P_303 != 0} the residual equations are triangular in ['G_220', 'G_112', 'G_120', 'G_030', 'G_021', 'G_012'] with diagonal (const)*P_303: a graph (irreducible)
  ok   R4 N2: on {P_303 = 0}: each equation is (P_203) times a linear form; components {c = 0} and {linear forms ['-2*G_220', '1/7*P_510 - G_112'] = 0}
  ok   R4 N2, component W1 (P_303 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.3 s, stabilization 0.1 s
  ok   R4 N2, component W1 (P_303 != 0) anchor (A): fixed {'P_303': 1}, solved ['G_220', 'G_112', 'G_120', 'G_030', 'G_021', 'G_012']; member closed with H and the prescribed graded data: True; a triple of I of local length 36 (ds), stabilization (36, 10)
      anchor generators: {'P': 'X^7 + X^3*Z^3 + 3/4*Z^4 + X*Y^2 + X*Y*Z + Y*Z^2 + Z^3', 'G': 'X^3*Z^2 + Z^3'}
      anchor triple: ['x^4*y^4 - 3*x^3*y^5 + x^2*y^6 + 2*y^8 - 2*x^7*z + 2*x^6*y*z + 3*x^5*y^2*z - 3*x^4*y^3*z - 3*x^3*y^4*z - 2*x^2*y^5*z + x*y^6*z + 2*y^7*z + 2*x^6*z^2 + 3*x^5*y*z^2 + x^4*y^2*z^2 + 3*x^3*y^3*z^2 - 2*x^2*y^4*z^2 - 3*x*y^5*z^2 - 3*y^6*z^2 - 3*x^5*z^3 - 2*x^4*y*z^3 - x^3*y^2*z^3 + 3*x^2*y^3*z^3 - x*y^4*z^3 + 2*y^5*z^3 - 2*x^4*z^4 - x^3*y*z^4 + 2*x^2*y^2*z^4 - 3*x*y^3*z^4 + 3*y^4*z^4 - 2*x^3*z^5 - 3*x^2*y*z^5 + x*y^2*z^5 - 2*y^3*z^5 - 3*x*y*z^6 + y^2*z^6 - x*z^7 - y*z^7 + 2*z^8 + 109/5040*x^7 + 3*x^6*y - 3*x^5*y^2 + x^4*y^3 - x^3*y^4 + x^2*y^5 - 3*x*y^6 + x^6*z - 3*x^5*y*z + 2*x^4*y^2*z - 3*x^3*y^3*z + 2*x*y^5*z + 3*x^5*z^2 + 2*x^4*y*z^2 - 3*x^3*y^2*z^2 + 3*x*y^4*z^2 - y^5*z^2 - x^4*z^3 + 2*x^3*y*z^3 - x^2*y^2*z^3 - 3*x*y^3*z^3 + y^4*z^3 - 2*x^3*z^4 + 2*x^2*y*z^4 - 3*x*y^2*z^4 - y^3*z^4 + 3*x^2*z^5 + 2*x*y*z^5 - y^2*z^5 + 2*y*z^6 + 3*z^7 - 3/140*x^6 - 2*x^4*y^2 - 3*x^3*y^3 + 2*x^2*y^4 + 3*y^6 - 2*x^5*z - 3*x^4*y*z + 3*y^5*z - x^4*z^2 + x^3*y*z^2 - 3*x^2*y^2*z^2 + 2*x*y^3*z^2 - 2*y^4*z^2 - 2*x^3*z^3 + x^2*y*z^3 + x*y^2*z^3 + x^2*z^4 + 2*x*y*z^4 - 2*x*z^5 + y*z^5 + 1/70*x^5 + 2*x^4*y - x^3*y^2 + 2*x*y^4 - 2*y^5 - 3*x^4*z - x^3*y*z - 2*x^2*y^2*z - x*y^3*z + y^4*z - 1/2*x^3*z^2 - x^2*y*z^2 - y^3*z^2 + 3*x^2*z^3 + 2*x*y*z^3 - 3*y^2*z^3 + 3*x*z^4 + 2*y*z^4 - 3*z^5 - 1/140*x^4 + x^3*y + x^2*y^2 + 3*x*y^3 + 3*y^4 - x*y^2*z + 2*x*y*z^2 - 2*y^2*z^2 - 2*x*z^3 + 2*y*z^3 - 2*z^4 + x*y^2 + 3*y^3 - 3*x*y*z - 3*y^2*z - 3*y*z^2 + z^3', '2*x^8 + 3*x^7*y - 3*x^6*y^2 + x^5*y^3 - 2*x*y^7 + y^8 - 3*x^7*z - 2*x^6*y*z + 2*x^5*y^2*z - x^4*y^3*z - 2*x^3*y^4*z - 3*x^2*y^5*z - 2*x*y^6*z - y^7*z + 2*x^6*z^2 - x^5*y*z^2 + x^4*y^2*z^2 - x^3*y^3*z^2 - 2*x^2*y^4*z^2 - 2*x*y^5*z^2 + y^6*z^2 + 2*x^5*z^3 - x^4*y*z^3 + 2*x^2*y^3*z^3 - 2*x*y^4*z^3 - 3*y^5*z^3 + 3*x^4*z^4 + 3*x^3*y*z^4 - 3*x^2*y^2*z^4 - x*y^3*z^4 - y^4*z^4 + x^3*z^5 + x^2*y*z^5 + 2*x*y^2*z^5 + 3*y^3*z^5 + 3*x^2*z^6 - x*y*z^6 - 2*y^2*z^6 + x*z^7 - 2*y*z^7 + z^8 + 131/5040*x^7 + x^6*y - 3*x^5*y^2 + 3*x^4*y^3 - 2*x^3*y^4 - x^2*y^5 + x*y^6 - y^7 - 2*x^6*z - x^5*y*z - 2*x^4*y^2*z - x^3*y^3*z - 3*x^2*y^4*z - 2*x*y^5*z + y^6*z - 2*x^5*z^2 + 2*x^4*y*z^2 + x^3*y^2*z^2 + 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - y^5*z^2 - x^4*z^3 - 2*x^3*y*z^3 + 3*x^2*y^2*z^3 + 2*x*y^3*z^3 - 3*y^4*z^3 - 2*x^3*z^4 - 2*x^2*y*z^4 - 2*x*y^2*z^4 + y^3*z^4 - 3*x^2*z^5 - 3*x*y*z^5 - 2*y^2*z^5 + x*z^6 - 3*z^7 + 3/140*x^6 - x^5*y + x^4*y^2 - 2*x^2*y^4 + x*y^5 - 2*y^6 - 3*x^5*z + 2*x^4*y*z + 3*x^3*y^2*z - x^2*y^3*z + x*y^4*z + y^5*z - 3*x^4*z^2 + x^3*y*z^2 + 2*x^2*y^2*z^2 - 3*x^3*z^3 + 2*x*y^2*z^3 - 2*x^2*z^4 - 2*x*y*z^4 - 3*y^2*z^4 - x*z^5 - 2*y*z^5 - z^6 + 1/140*x^5 + x^4*y + x^3*y^2 + 2*x^2*y^3 - x*y^4 + y^5 - x^4*z + 2*x^3*y*z + 3*x^2*y^2*z + 3*x*y^3*z - 2*y^4*z + x^3*z^2 - 3*x*y^2*z^2 + y^3*z^2 - 3*x^2*z^3 - x*y*z^3 + 3*x*z^4 + 3*y*z^4 - 2*z^5 + 1/70*x^4 - x^3*y + 2*x^2*y^2 + 3*x*y^3 + x^2*y*z + 2*x*y^2*z - 2*y^3*z + 2*x*y*z^2 - 2*y^2*z^2 - x*z^3 + y*z^3 - 3*x*y^2 - 3*y^3 - 3*x*y*z + 2*y^2*z - y*z^2 - 2*z^3', 'x^8 + 2*x^7*y + 3*x^6*y^2 + x^5*y^3 + 2*x^4*y^4 - 3*x^3*y^5 + x^2*y^6 + 2*x*y^7 + y^8 + 3*x^7*z - x^6*y*z - 2*x^5*y^2*z + 2*x^4*y^3*z - 2*x^3*y^4*z + 2*x^2*y^5*z + 2*x*y^6*z + 2*y^7*z - 2*x^4*y^2*z^2 + x^3*y^3*z^2 - 3*x^2*y^4*z^2 + x*y^5*z^2 - 3*y^6*z^2 - x^5*z^3 + 2*x^4*y*z^3 - 2*x^3*y^2*z^3 - 2*x^2*y^3*z^3 - 3*x*y^4*z^3 + 3*y^5*z^3 - 3*x^4*z^4 - x^3*y*z^4 + 3*x^2*y^2*z^4 + 3*x*y^3*z^4 - y^4*z^4 + x^3*z^5 - 2*x^2*y*z^5 + 2*y^3*z^5 - 2*x^2*z^6 + y^2*z^6 - 3*x*z^7 + 3*y*z^7 - 2*z^8 + 11/720*x^7 - x^5*y^2 + 3*x^3*y^4 - x^2*y^5 + 2*x*y^6 + 3*y^7 - 2*x^6*z + 3*x^5*y*z - x^4*y^2*z - 2*x^3*y^3*z - x^2*y^4*z + x*y^5*z - 2*y^6*z - x^5*z^2 - x^4*y*z^2 - 2*x^3*y^2*z^2 + 2*y^5*z^2 - 2*x^4*z^3 + 2*x^3*y*z^3 + 2*x^2*y^2*z^3 - 3*x*y^3*z^3 - 3*y^4*z^3 + x^3*z^4 + 3*x^2*y*z^4 - x*y^2*z^4 - 3*y^3*z^4 - x^2*z^5 + 3*x*z^6 - y*z^6 + 3*z^7 - 3*x^5*y + x^4*y^2 + 3*x^3*y^3 + x*y^5 - 2*x^5*z + 3*x^4*y*z - x^2*y^3*z + 2*x*y^4*z + y^5*z - 3*x^4*z^2 + 3*x^3*y*z^2 - 2*x^2*y^2*z^2 - 3*y^4*z^2 - 3*x^3*z^3 - 2*x^2*y*z^3 + 2*y^3*z^3 - 3*x*y*z^4 + y^2*z^4 - 3*x*z^5 - 3/140*x^5 - x^4*y - 2*x^3*y^2 + x^2*y^3 - y^5 - 3*x^4*z - x^3*y*z + x^2*y^2*z + 3*x*y^3*z - 3*y^4*z + x^3*z^2 + 3*x^2*y*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 - x*y*z^3 - 3*y^2*z^3 + 2*x*z^4 - 2*y*z^4 + 1/70*x^4 + 3*x^3*y - x*y^3 - 2*x^2*y*z + x*y^2*z - y^2*z^2 + 3*x*z^3 + 2*y*z^3 + 2*z^4 + x*y*z + 3*y^2*z + 3*y*z^2 - 2*z^3']
  ok   R4 N2, component W2 (P_303 = 0 = P_203): a point of the component (residual equations vanish)
      timing: member+triples 0.6 s, stabilization 0.3 s
  ok   R4 N2, component W2 (P_303 = 0 = P_203) anchor (A): fixed {'P_303': 0}, solved ['P_203']; member closed with H and the prescribed graded data: True; a triple of I of local length 39 (ds), stabilization (39, 13)
      anchor generators: {'P': 'X^7 - 2*Y^3 - X*Y*Z + 2*Y^2*Z + Y*Z^2 + 2*Z^3', 'G': 'X^3*Z^2 + X^2*Y^2 + X*Y*Z^2 - 2*X*Y^2 - 2*Y^3 + X*Y*Z + Y^2*Z + Y*Z^2 - Z^3'}
      anchor triple: ['-3*x^8 + 3*x^7*y + 2*x^6*y^2 + x^5*y^3 + x^4*y^4 + 2*x^3*y^5 - 2*x^2*y^6 - 3*x*y^7 - y^8 - x^7*z + x^5*y^2*z - 3*x^4*y^3*z + x^3*y^4*z - 2*x^2*y^5*z + 3*x^6*z^2 - x^5*y*z^2 + x^4*y^2*z^2 - x^3*y^3*z^2 - x*y^5*z^2 + 3*y^6*z^2 + 2*x^5*z^3 - 3*x^4*y*z^3 - 2*x^2*y^3*z^3 - 3*x*y^4*z^3 + y^5*z^3 + x^4*z^4 + 2*x^2*y^2*z^4 - 2*x*y^3*z^4 - x^3*z^5 - 3*x*y^2*z^5 - 3*y^3*z^5 + 2*x^2*z^6 + x*y*z^6 + 2*y^2*z^6 - 3*x*z^7 - 2*y*z^7 - 3*z^8 + 17/2520*x^7 - 2*x^5*y^2 - 2*x^4*y^3 - 3*x^3*y^4 + 2*x*y^6 + y^7 - x^6*z + 3*x^4*y^2*z + 3*x^3*y^3*z + 3*x^2*y^4*z - 3*y^6*z + 2*x^5*z^2 + 2*x^4*y*z^2 + 3*x^3*y^2*z^2 - x^2*y^3*z^2 + 3*x*y^4*z^2 + x^4*z^3 + 2*x^3*y*z^3 - x*y^3*z^3 + 2*y^4*z^3 + 2*x^3*z^4 + 3*x*y^2*z^4 + 3*y^3*z^4 + x^2*z^5 - 3*x*y*z^5 - 2*y^2*z^5 - 2*x*z^6 + 2*y*z^6 + 3*z^7 - x^5*y - 2*x^3*y^3 - x*y^5 - 2*y^6 - 3*x^5*z + x^4*y*z - 2*x^3*y^2*z + x^2*y^3*z + x*y^4*z + 3*y^5*z - x^4*z^2 - 2*x^3*y*z^2 - 3*x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 - x^2*y*z^3 + y^3*z^3 + x^2*z^4 + 3*y^2*z^4 - 3*x*z^5 - 3*z^6 + 3*x^4*y + 3*x^3*y^2 - x^2*y^3 - 3*x*y^4 - 2*x^4*z + 3*x^3*y*z + 3*x^2*y^2*z + x*y^3*z + 2*y^4*z + 2/3*x^3*z^2 - x^2*y*z^2 + x*y^2*z^2 + 2*y^3*z^2 - x^2*z^3 - 3*x*y*z^3 - y^2*z^3 - x*z^4 + 2*y*z^4 + 3*z^5 + x^3*y + 3*x*y^3 - y^4 - 2*x^2*y*z - 1/2*x^2*z^2 + 3*x*y*z^2 + y^2*z^2 + x*z^3 + y*z^3 - z^4 + 2*y^3 - y^2*z + 3*y*z^2 - z^3', '-3*x^8 - 2*x^7*y + x^6*y^2 - 2*x^5*y^3 - 3*x^4*y^4 - x^3*y^5 - 3*x^2*y^6 + x*y^7 - y^8 - x^7*z - 3*x^6*y*z - 2*x^5*y^2*z + 2*x^4*y^3*z - 3*x^3*y^4*z + 2*x^2*y^5*z - 3*x*y^6*z + 3*y^7*z - 2*x^5*y*z^2 - 3*x^4*y^2*z^2 - 2*x^3*y^3*z^2 - x^2*y^4*z^2 + 3*x*y^5*z^2 + y^6*z^2 + 3*x^5*z^3 - x^4*y*z^3 + 3*x^3*y^2*z^3 + x^2*y^3*z^3 - 2*x*y^4*z^3 + 3*y^5*z^3 - x^4*z^4 + 3*x^3*y*z^4 + 3*x^2*y^2*z^4 - x*y^3*z^4 + 2*y^4*z^4 - 2*x^3*z^5 - 3*x^2*y*z^5 + 2*x*y^2*z^5 - x^2*z^6 + 2*y^2*z^6 + 2*x*z^7 + 2*z^8 + 1/144*x^7 - x^6*y - 3*x^5*y^2 - x^3*y^4 + 3*x^2*y^5 - 3*x*y^6 - 3*y^7 + 3*x^6*z - 2*x^3*y^3*z - 2*x^2*y^4*z - x*y^5*z + 3*x^5*z^2 + x^4*y*z^2 - 3*x^3*y^2*z^2 + 2*x^2*y^3*z^2 - 2*x^4*z^3 + 2*x^3*y*z^3 - x^2*y^2*z^3 + y^4*z^3 + 2*x^3*z^4 - x*y^2*z^4 + 2*y^3*z^4 - 3*x^2*z^5 + x*y*z^5 + 3*y^2*z^5 - 3*x*z^6 - 2*y*z^6 + 2*z^7 - 2*x^5*y - x^4*y^2 - x^3*y^3 + 2*x^2*y^4 - 2*x*y^5 - y^6 - 3*x^4*y*z + 3*x^3*y^2*z + x^2*y^3*z + 2*x*y^4*z + y^5*z - x^4*z^2 + 3*x^3*y*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 + 3*y^4*z^2 + x^3*z^3 + x^2*y*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 + y^2*z^4 - 3*x*z^5 + 2*x^4*y - 2*x^3*y^2 + 2*x*y^4 - 2*y^5 - x^4*z - 2*x^3*y*z - x*y^3*z + 3/4*x^3*z^2 - 3*x^2*y*z^2 - 2*x*y^2*z^2 + y^3*z^2 - 2*x^2*z^3 + 3*x*y*z^3 - 2*y^2*z^3 - 2*x*z^4 + y*z^4 - z^5 - 2*x^3*y + 2*x^2*y^2 - 2*x*y^3 + 3*y^4 + 1/2*x^3*z - 3*x*y^2*z + 2*y^3*z + 1/2*x^2*z^2 - 2*x*y*z^2 - 2*x*z^3 - 2*y*z^3 - x*y^2 + y^3 - 3*x*y*z - 3*y^2*z - y*z^2 - z^3', '-x^8 - 2*x^7*y - 2*x^6*y^2 + 3*x^5*y^3 - 2*x^4*y^4 - 2*x^3*y^5 + x^2*y^6 - 2*x*y^7 + y^8 + x^6*y*z + x^5*y^2*z - x^4*y^3*z + 2*x^3*y^4*z + 2*x^2*y^5*z - 2*x*y^6*z - 3*y^7*z - 3*x^6*z^2 - x^5*y*z^2 - 3*x^4*y^2*z^2 - x^3*y^3*z^2 - 3*x^2*y^4*z^2 + x*y^5*z^2 + y^6*z^2 - 2*x^5*z^3 - 3*x^4*y*z^3 - x^3*y^2*z^3 - 2*x^2*y^3*z^3 - 3*x*y^4*z^3 + 2*y^5*z^3 - x^4*z^4 - 2*x^3*y*z^4 + 2*x*y^3*z^4 + 3*x^2*y*z^5 + x*y^2*z^5 + 2*y^3*z^5 - 3*x^2*z^6 + x*y*z^6 - 2*y^2*z^6 + 2*x*z^7 - 2*y*z^7 + 11/1680*x^7 + 2*x^6*y - 3*x^5*y^2 + 2*x^4*y^3 - 2*x^3*y^4 + 3*x^2*y^5 - x*y^6 + 2*y^7 - 3*x^6*z + x^5*y*z - 3*x^4*y^2*z - 2*x^3*y^3*z + x^2*y^4*z + x*y^5*z - 3*y^6*z + 3*x^5*z^2 + x^4*y*z^2 - 2*x^3*y^2*z^2 + x^2*y^3*z^2 + x*y^4*z^2 + 3*y^5*z^2 - 2*x^4*z^3 - 2*x^2*y^2*z^3 - x*y^3*z^3 - 2*y^4*z^3 + 2*x^3*z^4 - x*y^2*z^4 - 3*y^3*z^4 - x^2*z^5 - 2*x*y*z^5 + y^2*z^5 + x*z^6 - 3*y*z^6 + 3*x^5*y - x^4*y^2 + 3*x^3*y^3 - y^6 + 3*x^5*z + 3*x^4*y*z + 3*x^3*y^2*z + x^2*y^3*z - 3*x*y^4*z - 3*y^5*z - x^4*z^2 - 3*x^2*y^2*z^2 + 2*y^4*z^2 - 3*x^3*z^3 - 3*x^2*y*z^3 + 2*x*y^2*z^3 - 2*y^3*z^3 + 2*x^2*z^4 + x*y*z^4 - y^2*z^4 - x*z^5 - y*z^5 - z^6 - 2*x^3*y^2 + x^2*y^3 + y^5 - 2*x^4*z - x^3*y*z - 3*x^2*y^2*z + 2*x*y^3*z + 23/12*x^3*z^2 - x*y^2*z^2 + 3*y^3*z^2 - 3*x^2*z^3 - 3*x*y*z^3 + 3*y^2*z^3 + 3*x*z^4 - 3*y*z^4 - z^5 + x^3*y - 3*x^2*y^2 + 3*x*y^3 - 2*y^4 + 1/2*x^3*z + 3*x^2*y*z + 3*x*y^2*z + 2*y^3*z - 2/3*x^2*z^2 + 3*x*y*z^2 + 3*y^2*z^2 + 3*x*z^3 - 3*y*z^3 + 3*z^4 + 2*x*y^2 + 2*y^3 - 3*x*y*z + 3*y^2*z - 2*z^3']
  ok   R4 N2, component W3 (P_303 = 0, linear): a point of the component (residual equations vanish)
      timing: member+triples 0.3 s, stabilization 0.1 s
  ok   R4 N2, component W3 (P_303 = 0, linear) anchor (A): fixed {'P_303': 0}, solved ['G_220', 'G_112']; member closed with H and the prescribed graded data: True; a triple of I of local length 39 (ds), stabilization (39, 11)
      anchor generators: {'P': 'X^7 + X^2*Z^3 + X*Y^2 - 2*Y^3 + X*Y*Z + 2*Y^2*Z - 2*Y*Z^2', 'G': 'X^3*Z^2 - 2*X*Y^2 - 2*X*Y*Z + 2*Y^2*Z - 2*Z^3'}
      anchor triple: ['-2*x^8 + x^7*y - 2*x^6*y^2 - 2*x^5*y^3 + 3*x^4*y^4 - 3*x^3*y^5 + 2*x^2*y^6 + 2*x*y^7 + y^8 - 3*x^7*z + x^6*y*z + 2*x^5*y^2*z - 3*x^4*y^3*z + 2*x^3*y^4*z - 2*x^2*y^5*z + x*y^6*z - 3*x^6*z^2 - 2*x^5*y*z^2 - x^4*y^2*z^2 + 3*x^3*y^3*z^2 + x^2*y^4*z^2 - x*y^5*z^2 + 3*y^6*z^2 - 2*x^5*z^3 - x^4*y*z^3 + 2*x^3*y^2*z^3 + 2*x^2*y^3*z^3 - x*y^4*z^3 + 2*x^4*z^4 + 3*x^3*y*z^4 + x^2*y^2*z^4 + x*y^3*z^4 + 2*y^4*z^4 - 2*x^3*z^5 - 3*x^2*y*z^5 + 3*y^3*z^5 - 3*x^2*z^6 + x*y*z^6 - y^2*z^6 - x*z^7 + y*z^7 - 3*z^8 + 13/5040*x^7 - 3*x^6*y + 3*x^5*y^2 + 2*x^4*y^3 - x^3*y^4 - 2*x^2*y^5 + 3*x*y^6 + 2*y^7 - x^6*z + x^5*y*z - 2*x^4*y^2*z + x^3*y^3*z - 3*x^2*y^4*z - 3*y^6*z + 2*x^5*z^2 - 3*x^4*y*z^2 + 3*x^3*y^2*z^2 - x*y^4*z^2 - y^5*z^2 + x^4*z^3 - x^3*y*z^3 - x^2*y^2*z^3 - x*y^3*z^3 - 3*y^4*z^3 - x^3*z^4 + 3*x^2*y*z^4 + y^3*z^4 + 2*x^2*z^5 + x*y*z^5 + 2*y^2*z^5 + 3*x*z^6 - 2*y*z^6 - z^7 + 1/140*x^6 - 2*x^5*y + 2*x^3*y^3 + 2*x^2*y^4 + 3*x*y^5 + 3*y^6 - x^5*z + 3*x^4*y*z - 2*x^2*y^3*z - x*y^4*z + y^5*z - 2*x^3*y*z^2 - 2*x*y^3*z^2 + 3*y^4*z^2 - x^3*z^3 - 3*x^2*y*z^3 + 3*y^3*z^3 + y^2*z^4 - 3*x*z^5 + y*z^5 - 2*z^6 + 2*x^4*y + 3*x^3*y^2 - 3*x^2*y^3 + 2*x*y^4 - 2*y^5 - 2*x^4*z - x^3*y*z - 2*x*y^3*z + 2*y^4*z + 5/6*x^3*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 - 2*x*y*z^3 - 3*y^2*z^3 + 3*x*z^4 - 2*z^5 - x^3*y + 3*x^2*y^2 + 3*y^4 - 2*x^2*y*z + x*y^2*z - 3*x*y*z^2 + 2*y^2*z^2 - 3*x*z^3 - 3*y*z^3 - z^4 - x*y^2 + x*y*z - 3*y^2*z', '3*x^8 + 2*x^7*y - 2*x^6*y^2 + x^5*y^3 + 3*x^4*y^4 + x*y^7 + 2*x^6*y*z - 3*x^5*y^2*z - 3*x^2*y^5*z - x*y^6*z - 3*y^7*z - x^6*z^2 + 2*x^5*y*z^2 + 2*x^4*y^2*z^2 - 2*x^3*y^3*z^2 + x^2*y^4*z^2 - 2*x*y^5*z^2 - 3*y^6*z^2 + 3*x^5*z^3 - x^3*y^2*z^3 - x^2*y^3*z^3 - 3*x*y^4*z^3 - 3*x^4*z^4 + 3*x^3*y*z^4 + x^2*y^2*z^4 + x*y^3*z^4 - y^4*z^4 + 3*x^3*z^5 + x*y^2*z^5 - 3*x^2*z^6 - 2*y^2*z^6 + 2*x*z^7 - y*z^7 + 2*z^8 + 13/1008*x^7 + x^6*y + 2*x^4*y^3 + 2*x^3*y^4 + 3*x^2*y^5 - y^7 - 2*x^6*z + 2*x^5*y*z - 2*x^4*y^2*z + x^3*y^3*z + x^2*y^4*z - 3*x*y^5*z - y^6*z - 3*x^5*z^2 - 2*x^4*y*z^2 - 2*x^3*y^2*z^2 - 2*x^2*y^3*z^2 - 3*y^5*z^2 + 3*x^4*z^3 - 2*x^3*y*z^3 + 3*x^2*y^2*z^3 - x*y^3*z^3 - 3*y^4*z^3 + x^3*z^4 + 3*x^2*y*z^4 - 3*x*y^2*z^4 + 2*y^3*z^4 - x^2*z^5 + 2*x*y*z^5 + 2*y^2*z^5 - 2*x*z^6 + 3*y*z^6 - 2*z^7 + 3*x^5*y - 2*x^4*y^2 + 2*x^3*y^3 + 3*x^2*y^4 + 3*x*y^5 + y^6 + 2*x^5*z - 3*x^4*y*z - 3*x^3*y^2*z + 3*x^2*y^3*z - x*y^4*z - y^5*z - x^4*z^2 + x^3*y*z^2 - 2*x^2*y^2*z^2 + x*y^3*z^2 + y^4*z^2 - x^3*z^3 + x^2*y*z^3 - 2*x*y^2*z^3 + y^3*z^3 - 2*x*y*z^4 - 3*y^2*z^4 + x*z^5 + 3*y*z^5 - 3*z^6 + 1/420*x^5 - 2*x^4*y - x^3*y^2 + 2*x*y^4 - x^4*z + 3*x^2*y^2*z + 2*x*y^3*z + 3*y^4*z - 1/6*x^3*z^2 + 3*x*y^2*z^2 + y^3*z^2 - 2*x^2*z^3 - x*z^4 - y*z^4 - z^5 - x^3*y + x^2*y^2 - x*y^3 - y^4 - 2*x^2*y*z + 3*x*y^2*z + y^3*z - x*y*z^2 + y^2*z^2 - 2*y*z^3 - 3*z^4 - x*y^2 + 2*y^3 + x*y*z - 3*y^2*z + y*z^2 - z^3', '2*x^6*y^2 + x^5*y^3 - 3*x^4*y^4 - 2*x^3*y^5 - y^8 - 2*x^7*z + 3*x^6*y*z + 3*x^5*y^2*z + x^3*y^4*z - 3*x^2*y^5*z - 2*x*y^6*z + 3*y^7*z + 2*x^5*y*z^2 - 3*x^4*y^2*z^2 + 3*x^2*y^4*z^2 - 3*x*y^5*z^2 + x^4*y*z^3 + 3*x^3*y^2*z^3 - 3*x^2*y^3*z^3 + 3*x*y^4*z^3 - 2*y^5*z^3 - 3*x^4*z^4 - x^3*y*z^4 - 2*x^2*y^2*z^4 + 3*x*y^3*z^4 + 3*y^4*z^4 - x^2*y*z^5 - 2*x*y^2*z^5 - 3*y^3*z^5 + 3*x^2*z^6 + 3*x*y*z^6 + 2*y^2*z^6 - x*z^7 - 2*z^8 + 11/840*x^7 + x^6*y + 3*x^5*y^2 - 2*x^4*y^3 - x*y^6 - 2*y^7 - 3*x^5*y*z - x^3*y^3*z - 3*x^2*y^4*z + 3*x*y^5*z + 2*y^6*z + 3*x^5*z^2 + 3*x^4*y*z^2 + x^3*y^2*z^2 - 3*x^2*y^3*z^2 - 2*x*y^4*z^2 + 3*y^5*z^2 + x^4*z^3 - 2*x^3*y*z^3 + 2*x^2*y^2*z^3 + 2*x*y^3*z^3 - 3*y^4*z^3 - 3*x^3*z^4 - x^2*y*z^4 - 2*x*y^2*z^4 - 2*y^3*z^4 - 3*x^2*z^5 - 3*x*y*z^5 - y^2*z^5 + 3*x*z^6 + y*z^6 + z^7 + 3*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - 3*y^6 + x^5*z - x^4*y*z - 3*x^3*y^2*z + 2*x^2*y^3*z - 2*x*y^4*z - y^5*z - 3*x^4*z^2 - x^3*y*z^2 - 3*x^2*y^2*z^2 - x*y^3*z^2 + y^4*z^2 + 3*x^3*z^3 + 2*x^2*y*z^3 + 3*y^3*z^3 - 2*x^2*z^4 + 2*x*y*z^4 + x*z^5 + 3*y*z^5 + z^6 + 1/420*x^5 + 3*x^3*y^2 - 2*x^2*y^3 - x*y^4 + y^5 - 3*x^4*z - x^3*y*z + 3*x^2*y^2*z - x*y^3*z - 2*y^4*z - x^3*z^2 - 2*x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 - 3*y^2*z^3 - x*z^4 - y*z^4 - 2*x^3*y - 3*x^2*y^2 + 2*x*y^3 - 2*y^4 + 3*x^2*y*z - x*y^2*z + 3*y^3*z - x*y*z^2 - 3*y^2*z^2 + 2*y*z^3 + z^4 - 2*x*y^2 + 3*y^3 - 2*x*y*z - 3*y^2*z - z^3']
--- R4 N3
  ok   R4 N3: the graded normal form is closed with H = [1, 3, 6, 4, 3, 2, 1, 1]
  ok   R4 N3: one lift per pivot of D_4..D_7
    134 closedness conditions in 88 tail coefficients; 56 eliminated with constant pivots; residual equations:
       (-2) * G_220 * P_402
       (1/42) * (6*P_510*P_402*G_310 + 10*P_501^2*G_220 + 21*P_402*G_310*G_211 + 12*P_501*P_402*H_120 - 6*P_510*P_402*H_111 + 21*P_402*G_211*H_111 - 42*P_302*G_220 - 84*P_402*G_120 - 126*P_203*H_120)
       (1/21) * (6*P_501*P_402*H_030 - 2*P_510*P_402*H_021 + 7*P_402*G_211*H_021 - 42*P_402*G_030 - 63*P_203*H_030)
       (1/21) * (P_510*P_402*G_211 + 6*P_501*P_402*H_021 - 6*P_510*P_402*H_012 + 21*P_402*G_211*H_012 - 42*P_402*G_021 - 63*P_203*H_021)
       (1/42) * (-2*P_501*P_402*G_211 + 63*P_402*G_310*G_103 + 12*P_501*P_402*H_012 - 36*P_510*P_402*H_003 + 126*P_402*G_211*H_003 + 6*P_510*P_203 - 84*P_402*G_012 - 126*P_203*H_012)
       (-4/7) * G_220 * P_402
  ok   R4 N3: on {P_402 != 0} the residual equations are triangular in ['G_220', 'G_120', 'G_030', 'G_021', 'G_012'] with diagonal (const)*P_402: a graph (irreducible)
  ok   R4 N3: on {P_402 = 0}: on {P_203 != 0} triangular in ['H_120', 'H_030', 'H_021', 'H_012'] with diagonal (const)*P_203; on {P_203 = 0}: multiples of 5*P_501^2*G_220 - 21*P_302*G_220: ['(-1/21) * G_220 * (-5*P_501^2 + 21*P_302)']
  ok   R4 N3, component W1 (P_402 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.3 s, stabilization 0.1 s
  ok   R4 N3, component W1 (P_402 != 0) anchor (A): fixed {'P_402': 1}, solved ['G_220', 'G_120', 'G_030', 'G_021', 'G_012']; member closed with H and the prescribed graded data: True; a triple of I of local length 37 (ds), stabilization (37, 11)
      anchor generators: {'P': 'X^7 + X^4*Z^2 + 1/2*Z^4 + 2*X*Y^2 + Y^3 + 2*X*Y*Z + Y^2*Z - 2*Z^3', 'G': 'X^4*Z + Z^3', 'H': 'X^2*Z^2 + Y^3 + X*Y*Z + 2*Y^2*Z + Y*Z^2 - Z^3'}
      anchor triple: ['x^8 + 2*x^7*y - x^6*y^2 + 2*x^5*y^3 - 2*x^4*y^4 + 3*x^3*y^5 + 3*x^2*y^6 - x*y^7 + 3*y^8 - 2*x^7*z + 2*x^6*y*z - 2*x^5*y^2*z - x^4*y^3*z + 2*x^3*y^4*z - 3*x^2*y^5*z - 3*x*y^6*z + 2*x^6*z^2 + 3*x^5*y*z^2 + x^4*y^2*z^2 + 2*x^3*y^3*z^2 + 2*x^2*y^4*z^2 - x*y^5*z^2 - 2*y^6*z^2 + 3*x^5*z^3 + x^4*y*z^3 - 2*x^3*y^2*z^3 + 2*x^2*y^3*z^3 + 2*x*y^4*z^3 - 3*y^5*z^3 - 2*x^3*y*z^4 - x^2*y^2*z^4 - x*y^3*z^4 - 3*y^4*z^4 - 3*x^3*z^5 + 2*x*y^2*z^5 + 2*y^3*z^5 + 2*x^2*z^6 - 2*x*y*z^6 + y^2*z^6 + x*z^7 - 2*y*z^7 + 13/840*x^7 - 3*x^6*y + x^4*y^3 - x^3*y^4 - 3*x^2*y^5 + x*y^6 + y^7 - 3*x^6*z + 2*x^5*y*z + 2*x^4*y^2*z - 3*x^3*y^3*z - x^2*y^4*z - x*y^5*z - 3*y^6*z - 3*x^5*z^2 + 3*x^4*y*z^2 + x^3*y^2*z^2 + 2*x^2*y^3*z^2 + 3*x*y^4*z^2 + 3*y^5*z^2 + 2*x^3*y*z^3 + 3*x*y^3*z^3 + 2*y^4*z^3 - 2*x^3*z^4 - x^2*y*z^4 - 3*x*y^2*z^4 + 2*y^3*z^4 - 2*x^2*z^5 - x*y*z^5 - 3*y^2*z^5 + x*z^6 + 2*y*z^6 + 2*z^7 + 2/105*x^6 - 2*x^5*y + x^4*y^2 - 2*x^3*y^3 - 3*x^2*y^4 + 3*x*y^5 - 2*y^6 - 2*x^5*z - x^3*y^2*z + x^2*y^3*z + 3*x*y^4*z - 2*y^5*z - x^4*z^2 + 3*x^3*y*z^2 - 3*x^2*y^2*z^2 - 2*x*y^3*z^2 + x^3*z^3 - x^2*y*z^3 + 2*x*y^2*z^3 - 3*y^3*z^3 - 3*x^2*z^4 - 2*y^2*z^4 + 2*x*z^5 - y*z^5 - z^6 - 31/420*x^5 - x^4*y + 3*x^3*y^2 + x^2*y^3 + 2*x*y^4 - y^5 - 3/4*x^4*z + 2*x^2*y^2*z - x*y^3*z - 2*x^3*z^2 - x^2*y*z^2 + x*y^2*z^2 - 2*y^3*z^2 + x^2*z^3 - 3*x*y*z^3 - y^2*z^3 + x*z^4 - 3*y*z^4 - z^5 - x^2*y^2 - 3*x*y^3 - 3*y^4 - x^2*y*z - 2*y^3*z + 31/4*x^2*z^2 - x*y*z^2 + 2*y^2*z^2 + 2*x*z^3 + 2*y*z^3 + 2*z^4 - 3*x*y^2 + y^3 - 3*x*y*z - 3*y^2*z - 2*y*z^2 + 3*z^3', 'x^7*y + 3*x^5*y^3 - 3*x^3*y^5 - 2*x^2*y^6 + 3*x*y^7 + 3*y^8 - 2*x^7*z + x^6*y*z - 3*x^5*y^2*z - 2*x^3*y^4*z + 3*x^2*y^5*z + 2*x*y^6*z + 3*y^7*z + 2*x^6*z^2 + 3*x^5*y*z^2 + 2*x^3*y^3*z^2 + x^2*y^4*z^2 - 2*y^6*z^2 + 3*x^5*z^3 + x^4*y*z^3 + 2*x^3*y^2*z^3 - 3*x*y^4*z^3 - y^5*z^3 - 2*x^4*z^4 - x^3*y*z^4 + x*y^3*z^4 + x^2*y*z^5 - 3*x*y^2*z^5 - 3*y^3*z^5 + 3*x^2*z^6 + 3*x*y*z^6 - 3*y^2*z^6 + x*z^7 + 2*y*z^7 + 3*z^8 + 23/840*x^7 - 3*x^6*y + x^5*y^2 - 3*x^4*y^3 - 2*x^3*y^4 + x^2*y^5 + 3*y^7 - 2*x^6*z + 2*x^5*y*z + 3*x^4*y^2*z + x^3*y^3*z + 2*x^2*y^4*z - x*y^5*z + 3*y^6*z - 3*x^5*z^2 + 3*x^4*y*z^2 + 3*x^3*y^2*z^2 - 2*x^2*y^3*z^2 + 2*x*y^4*z^2 - y^5*z^2 + x^4*z^3 - x^3*y*z^3 - 2*x*y^3*z^3 - y^4*z^3 + 3*x^3*z^4 - 2*x^2*y*z^4 - x*y^2*z^4 + y^3*z^4 + 2*x^2*z^5 + 2*x*y*z^5 - y^2*z^5 - x*z^6 + 3*y*z^6 - 2*z^7 + 1/105*x^6 + x^5*y + 2*x^4*y^2 + 3*x^3*y^3 + 2*x^2*y^4 - x*y^5 - y^6 - x^5*z + 3*x^4*y*z + 2*x^3*y^2*z + 3*x^2*y^3*z - x*y^4*z - 2*y^5*z - 3*x^4*z^2 - 2*x^3*y*z^2 + x^2*y^2*z^2 - x*y^3*z^2 + 2*y^4*z^2 + 3*x^3*z^3 + 2*x*y^2*z^3 - 3*y^3*z^3 + 2*x*y*z^4 - y^2*z^4 - 2*x*z^5 + y*z^5 + 2*z^6 - 1/140*x^5 + 3*x^4*y - 3*x^3*y^2 + x*y^4 + 3*y^5 - 1/2*x^4*z - 2*x^2*y^2*z - 2*x*y^3*z - y^4*z - x^3*z^2 + 2*x^2*y*z^2 + x*y^2*z^2 + 3*y^3*z^2 - 2*x^2*z^3 + 2*x*y*z^3 - 3*y^2*z^3 - y*z^4 + 3*z^5 - 3*x^3*y + 2*x^2*y^2 + 2*x*y^3 + 2*y^4 - 3*x^2*y*z - 3*x*y^2*z + 2*y^3*z + 3/4*x^2*z^2 + x*y*z^2 + 2*y^2*z^2 + 3*x*z^3 + 2*y*z^3 + 2*z^4 + 3*x*y*z + 3*y*z^2 + 2*z^3', '-2*x^8 - 2*x^7*y + 3*x^4*y^4 - 3*x^3*y^5 - x^2*y^6 - 2*x*y^7 + y^8 + x^7*z + 2*x^6*y*z - 2*x^5*y^2*z - 2*x^4*y^3*z - x^3*y^4*z - 3*x^2*y^5*z + x*y^6*z - 3*y^7*z + x^6*z^2 - 2*x^5*y*z^2 - 2*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + 3*x^2*y^4*z^2 + x*y^5*z^2 + 2*y^6*z^2 - 3*x^5*z^3 + 2*x^4*y*z^3 + x^3*y^2*z^3 - x^2*y^3*z^3 + 3*x*y^4*z^3 + y^5*z^3 - 2*x^4*z^4 - 3*x^3*y*z^4 + x^2*y^2*z^4 + 3*x*y^3*z^4 + y^4*z^4 + 2*x^3*z^5 - 2*x^2*y*z^5 + 2*x*y^2*z^5 - 2*y^3*z^5 + 3*x^2*z^6 - 2*x*y*z^6 - y^2*z^6 - 3*x*z^7 + y*z^7 - 29/1260*x^7 - 3*x^5*y^2 + 3*x^4*y^3 - 2*x^3*y^4 + 3*x*y^6 - 3*y^7 - 2*x^6*z - x^5*y*z + 2*x^4*y^2*z - 3*x^3*y^3*z + 3*x^2*y^4*z + 3*x*y^5*z - 3*y^6*z + 2*x^5*z^2 - x^4*y*z^2 + 3*x^3*y^2*z^2 - 2*x^2*y^3*z^2 + 3*y^5*z^2 - x^4*z^3 - 3*x^3*y*z^3 - x*y^3*z^3 + x^3*z^4 - 2*x^2*y*z^4 + 2*x*y^2*z^4 - 2*y^3*z^4 - 3*x^2*z^5 - 2*x*y*z^5 + 3*y^2*z^5 + 3*x*z^6 - 2*z^7 - 1/35*x^6 - 3*x^5*y + x^4*y^2 - 2*x^3*y^3 - x^2*y^4 + x*y^5 + 3*y^6 - x^4*y*z - 3*x^2*y^3*z + 3*x*y^4*z - 3*y^5*z + 3*x^4*z^2 - 3*x*y^3*z^2 - 3*y^4*z^2 - 2*x^3*z^3 - 3*x*y^2*z^3 + 2*y^3*z^3 + 2*x^2*z^4 - 2*x*y*z^4 + 2*x*z^5 - 3*y*z^5 + 3*z^6 + 1/210*x^5 - 3*x^4*y + x^3*y^2 + 2*x^2*y^3 - 3*x*y^4 - 1/2*x^4*z + x^3*y*z - 3*x^2*y^2*z + 2*x*y^3*z - y^4*z + 3*x^3*z^2 - 3*y^3*z^2 - x^2*z^3 + 3*x*y*z^3 + 3*y^2*z^3 + x*z^4 - 2*y*z^4 + z^5 + x^3*y + 3*x^2*y^2 + y^4 + 2*x^2*y*z - 3*x*y^2*z - 3*y^3*z - 1/2*x^2*z^2 - 3*x*y*z^2 + 3*y^2*z^2 - 2*x*z^3 - 3*y*z^3 - 2*z^4 + 3*x*y^2 + y^3 - 2*x*y*z + 3*y^2*z - y*z^2 + 2*z^3']
  ok   R4 N3, component W2 (P_402 = 0, P_203 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N3, component W2 (P_402 = 0, P_203 != 0) anchor (A): fixed {'P_402': 0, 'P_203': 1}, solved ['H_120', 'H_030', 'H_021', 'H_012']; member closed with H and the prescribed graded data: True; a triple of I of local length 39 (ds), stabilization (39, 11)
      anchor generators: {'P': 'X^7 + X^2*Z^3 + 3/4*Z^4 + X*Y^2 - 2*Y^3 - 2*X*Y*Z + Y^2*Z - 2*Z^3', 'G': 'X^4*Z - 2*X*Y^2 - 2*Y^3 - 2*X*Y*Z + 2*Y^2*Z - 2*Y*Z^2 + 2*Z^3', 'H': 'X^2*Z^2 + Z^3'}
      anchor triple: ['2*x^7*y + x^6*y^2 + x^5*y^3 + 2*x^4*y^4 + 2*x^3*y^5 + x^2*y^6 - 3*x*y^7 + y^8 + 3*x^6*y*z - 3*x^5*y^2*z + 2*x^4*y^3*z + 2*x^3*y^4*z + 2*x^2*y^5*z + 3*x*y^6*z + 2*y^7*z + x^6*z^2 + 3*x^5*y*z^2 - 3*x^3*y^3*z^2 - 2*x*y^5*z^2 + y^6*z^2 - 3*x^5*z^3 + 2*x^4*y*z^3 - 3*x^3*y^2*z^3 + x*y^4*z^3 - 3*y^5*z^3 + x^4*z^4 + x^3*y*z^4 - 3*x*y^3*z^4 - 3*x^3*z^5 - 3*x^2*y*z^5 + x*y^2*z^5 + y^3*z^5 - 2*x^2*z^6 + 3*x*z^7 - z^8 - 1/168*x^7 + 2*x^6*y + x^5*y^2 + 2*x^3*y^4 + 2*x^2*y^5 - 2*x*y^6 + y^7 - 3*x^6*z + x^4*y^2*z + 3*x^3*y^3*z + x^2*y^4*z - x*y^5*z + 3*y^6*z - 2*x^5*z^2 - 2*x^4*y*z^2 + 2*x^3*y^2*z^2 - 3*x^2*y^3*z^2 - x*y^4*z^2 + 3*y^5*z^2 + 3*x^4*z^3 + 2*x^3*y*z^3 - 3*x^2*y^2*z^3 - 3*x*y^3*z^3 + y^4*z^3 - 2*x^3*z^4 + 2*x^2*y*z^4 - 3*x*y^2*z^4 + x^2*z^5 + 2*y^2*z^5 - x*z^6 - y*z^6 - z^7 - 1/210*x^6 + 3*x^4*y^2 + 3*x^2*y^4 - 2*y^6 - x^4*y*z + 3*x^3*y^2*z - 2*x*y^4*z - x^4*z^2 + 3*x^2*y^2*z^2 - 3*x*y^3*z^2 - y^4*z^2 + x^3*z^3 + 2*x^2*y*z^3 + 3*x*y^2*z^3 + y^3*z^3 + x^2*z^4 - 2*x*y*z^4 + y^2*z^4 - 3*x*z^5 + y*z^5 + 1/210*x^5 - 3*x^4*y + 2*x^3*y^2 + 2*x^2*y^3 + x*y^4 + 3*y^5 + 3/4*x^4*z - 3*x^3*y*z - 2*x^2*y^2*z + 3*y^4*z - x^3*z^2 - 3*x^2*y*z^2 - 3*x*y^2*z^2 + 3*x^2*z^3 - 2*x*y*z^3 + y^2*z^3 + x*z^4 - y*z^4 + 2*z^5 - x^3*y + 2*x^2*y^2 - x*y^3 - y^4 + 2*x^2*y*z - 3*x*y^2*z + 3*y^3*z + 3*x^2*z^2 + 2*x*y*z^2 + 2*y^2*z^2 + 2*x*z^3 + y*z^3 - z^4 - 2*x*y^2 + x*y*z - 3*y^2*z - 3*y*z^2 - 2*z^3', '2*x^8 - 2*x^7*y - x^6*y^2 - 2*x^5*y^3 - 2*x^4*y^4 - x^3*y^5 - x^2*y^6 - 3*x*y^7 + 3*y^8 + 2*x^7*z - 3*x^6*y*z - 2*x^5*y^2*z + 3*x^3*y^4*z - 3*x^2*y^5*z + x*y^6*z + 3*y^7*z + x^6*z^2 + 3*x^5*y*z^2 - 2*x^4*y^2*z^2 + 3*x^2*y^4*z^2 + 2*x*y^5*z^2 + y^6*z^2 - 3*x^5*z^3 + 2*x^4*y*z^3 - 3*x^3*y^2*z^3 - 3*x^2*y^3*z^3 - x*y^4*z^3 + y^5*z^3 - 2*x^4*z^4 - 3*x^2*y^2*z^4 - x*y^3*z^4 - y^4*z^4 - x^3*z^5 + 3*x^2*y*z^5 - 2*x*y^2*z^5 - 2*y^2*z^6 + 2*x*z^7 - y*z^7 - 1/72*x^7 + 3*x^5*y^2 + 2*x^4*y^3 + 3*x^3*y^4 - x^2*y^5 - 2*x*y^6 - y^7 - 3*x^6*z + x^5*y*z + 3*x^4*y^2*z + x^3*y^3*z + 3*x^2*y^4*z + 2*x*y^5*z - 2*y^6*z - 2*x^5*z^2 + x^4*y*z^2 + 2*x^3*y^2*z^2 - x*y^4*z^2 - y^5*z^2 - 3*x^3*y*z^3 - 3*x*y^3*z^3 - y^4*z^3 - 3*x^3*z^4 + x*y^2*z^4 - 3*x^2*z^5 + x*z^6 - 2*y*z^6 + 2*z^7 + 1/210*x^6 + x^5*y + 3*x^4*y^2 + 3*x^3*y^3 + 3*x*y^5 - 3*y^6 + 2*x^5*z + 2*x^4*y*z - 2*x^3*y^2*z + x^2*y^3*z - x*y^4*z + 3*y^5*z + 3*x^4*z^2 - 3*x^3*y*z^2 + 2*x^2*y^2*z^2 + 3*x*y^3*z^2 + 3*y^4*z^2 - x^3*z^3 - x^2*y*z^3 + 3*x*y^2*z^3 + 3*y^3*z^3 + x^2*z^4 + 3*x*y*z^4 + 2*y^2*z^4 - 2*x*z^5 - 2*y*z^5 + z^6 + 1/140*x^5 - x^4*y - 2*x^3*y^2 + x^2*y^3 + 2*x*y^4 + 1/3*x^4*z + 2*x^3*y*z + 3*x^2*y^2*z - 2*x*y^3*z - 2*y^4*z + 3*x^3*z^2 + 3*x^2*y*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 - 3*x^2*z^3 + 2*x*y*z^3 + 3*y^2*z^3 + 2*x*z^4 + 2*y*z^4 - z^5 - 2*x^3*y - x^2*y^2 + 3*x*y^3 - 2*y^4 + 2*x^2*y*z + x*y^2*z - 3*y^3*z + 9/2*x^2*z^2 - 2*x*y*z^2 + 2*y^2*z^2 - 2*x*z^3 + 2*y*z^3 + 3*z^4 - 2*x*y^2 - y^3 - 2*x*y*z + 2*y^2*z + y*z^2 - 3*z^3', '2*x^8 - 2*x^7*y + 2*x^6*y^2 + 2*x^4*y^4 + 2*x^3*y^5 + 2*x^2*y^6 + 3*y^8 + 3*x^7*z - 3*x^6*y*z - x^5*y^2*z - 2*x^4*y^3*z - x^3*y^4*z + 3*x^2*y^5*z + x*y^6*z + 3*y^7*z + 2*x^6*z^2 - x^5*y*z^2 - 3*x^4*y^2*z^2 - 2*x^3*y^3*z^2 + 3*x^2*y^4*z^2 + 2*x*y^5*z^2 + 2*y^6*z^2 - 3*x^4*y*z^3 + x^3*y^2*z^3 + 2*x^2*y^3*z^3 - 2*x*y^4*z^3 - 3*y^5*z^3 - 3*x^4*z^4 + 2*x^3*y*z^4 - 3*x^2*y^2*z^4 - 3*x*y^3*z^4 - 3*y^4*z^4 + x^3*z^5 - x^2*y*z^5 + x*y^2*z^5 + 2*y^3*z^5 - 3*x^2*z^6 + x*y*z^6 + 3*y^2*z^6 + 3*x*z^7 + 3*y*z^7 - 1/60*x^7 - 2*x^6*y + x^5*y^2 - 3*x^4*y^3 + 3*x^3*y^4 + x^2*y^5 + 3*x*y^6 - 3*y^7 + 2*x^6*z + 3*x^5*y*z + x^4*y^2*z - 3*x^3*y^3*z - x^2*y^4*z + 2*x*y^5*z + 2*y^6*z - 3*x^5*z^2 - 2*x^4*y*z^2 + x^3*y^2*z^2 - 2*x*y^4*z^2 + 2*y^5*z^2 + x^4*z^3 - x^3*y*z^3 - 3*x^2*y^2*z^3 + 3*x*y^3*z^3 - y^4*z^3 + 3*x^3*z^4 + 3*x*y^2*z^4 - y^3*z^4 + 3*x^2*z^5 + 3*x*y*z^5 - x*z^6 - 2*y*z^6 - 2*z^7 - 1/420*x^6 + 3*x^3*y^3 + 3*x^2*y^4 - 3*x*y^5 - 3*y^6 - 2*x^5*z - 3*x^4*y*z + 2*x^3*y^2*z + 2*x^2*y^3*z + 2*x*y^4*z - y^5*z - x^4*z^2 + 2*x^3*y*z^2 + 3*x^2*y^2*z^2 + x*y^3*z^2 + y^4*z^2 + 3*x^3*z^3 - 2*x^2*y*z^3 + 3*x*y^2*z^3 - y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - 2*y^2*z^4 + 2*x*z^5 + 3*y*z^5 + z^6 + 1/420*x^5 - 2*x^4*y - x^3*y^2 + x^2*y^3 - x*y^4 + y^5 + 4/3*x^4*z - 2*x^3*y*z - 3*x^2*y^2*z - 2*x*y^3*z + 3*y^4*z - x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 - 2*x*y*z^3 - 3*x*z^4 + 3*y*z^4 + 3*z^5 + 3*x^3*y + 3*x*y^3 - 3*y^4 - 2*x^2*y*z - x*y^2*z - 3*y^3*z + 3/2*x^2*z^2 + x*z^3 + 2*y*z^3 + 3*z^4 + 3*x*y^2 - 2*x*y*z - 2*y^2*z + y*z^2 - z^3']
  ok   R4 N3, component W3 (P_402 = P_203 = 0, G_220 = 0): a point of the component (residual equations vanish)
      timing: member+triples 0.5 s, stabilization 0.2 s
  ok   R4 N3, component W3 (P_402 = P_203 = 0, G_220 = 0) anchor (A): fixed {'P_402': 0, 'P_203': 0, 'G_220': 0}, solved []; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 12)
      anchor generators: {'P': 'X^7 + 2*X^5*Y + 2*X^5*Z - X^4*Y + 20/21*X^3*Y^2 + 40/21*X^3*Y*Z - X^3*Y - 2*X^2*Y^2 + 40/441*X*Y^3 - 2*X^2*Y*Z + 40/147*X*Y^2*Z - 100/147*X*Y*Z^2 - X*Z^3 - 5/21*Z^4 + X*Y^2 - Y^3 + X*Y*Z + Y^2*Z - 2*Y*Z^2 + 2*Z^3', 'G': 'X^4*Z + X^3*Y + 2*X^2*Y*Z + 2*X*Z^3 - 2*Y^2*Z + Y*Z^2 - 2*Z^3', 'H': 'X^2*Z^2 + X*Y^2 + 2*X*Y*Z - Y^2*Z + 2*Y*Z^2 - 2*Z^3'}
      anchor triple: ['x^8 + 2*x^7*y - 2*x^5*y^3 - 3*x^4*y^4 - x^3*y^5 + x^2*y^6 - x*y^7 + 2*y^8 + 3*x^7*z - 3*x^5*y^2*z - x^4*y^3*z - x*y^6*z + y^7*z - 2*x^6*z^2 + x^4*y^2*z^2 - 3*x^3*y^3*z^2 + x^2*y^4*z^2 - 2*x*y^5*z^2 + 3*y^6*z^2 + 3*x^5*z^3 + 3*x^4*y*z^3 - 3*x^3*y^2*z^3 + 2*x^2*y^3*z^3 - x*y^4*z^3 + 3*y^5*z^3 - x^4*z^4 - x^3*y*z^4 - 2*x^2*y^2*z^4 + x*y^3*z^4 - 2*y^4*z^4 - 2*x^2*y*z^5 - 2*x*y^2*z^5 - y^3*z^5 + x^2*z^6 + x*y*z^6 + 2*y^2*z^6 - x*z^7 + 3*y*z^7 - z^8 - 349/30870*x^7 - 3*x^6*y + 2*x^4*y^3 + 2*x^3*y^4 + x^2*y^5 - 3*x*y^6 - 2*y^7 - 2*x^6*z - x^5*y*z + 3*x^4*y^2*z + 2*x^3*y^3*z + 3*x*y^5*z - 3*y^6*z + 2*x^5*z^2 + 3*x^4*y*z^2 - 2*x^3*y^2*z^2 + 3*x^2*y^3*z^2 + 2*y^5*z^2 + x^4*z^3 + 3*x^3*y*z^3 + 3*x^2*y^2*z^3 - 2*x*y^3*z^3 - 2*y^4*z^3 - x^3*z^4 + 2*x^2*y*z^4 - 3*y^3*z^4 + x^2*z^5 - 2*x*y*z^5 + x*z^6 + 2*y*z^6 - 2*z^7 + 9577/52920*x^6 - x^5*y + 2*x^4*y^2 - 3*x^3*y^3 + x^2*y^4 + 3*x*y^5 + x^5*z + x^3*y^2*z + 3*x^2*y^3*z + 3*y^5*z - 2*x^4*z^2 + x^3*y*z^2 - 3*x^2*y^2*z^2 - 3*x*y^3*z^2 - 2*x^3*z^3 - 3*x^2*y*z^3 - 3*x*y^2*z^3 + 3*y^3*z^3 + 2*x^2*z^4 + x*y*z^4 + y^2*z^4 - 3*x*z^5 - 3*y*z^5 + z^6 - 13/126*x^5 - 3*x^4*y - 2*x^3*y^2 + 3*x^2*y^3 - x*y^4 - 2*y^5 - 1/2*x^4*z - 2*x^2*y^2*z + 3*x*y^3*z - y^4*z - x^3*z^2 + 2*x^2*y*z^2 + x^2*z^3 - 3*x*y*z^3 - 2*y^2*z^3 + x*z^4 - 3*y*z^4 - 3*z^5 + 2*x^3*y - 3*x^2*y^2 + x*y^3 + y^4 + 1/6*x^3*z + 2*x*y^2*z - y^3*z - 9/2*x^2*z^2 + 3*x*y*z^2 - 3*y^2*z^2 - x*z^3 + y*z^3 + z^4 - 2*x*y^2 - 3*y^3 + 2*x*y*z + y^2*z + 2*y*z^2 - z^3', '2*x^8 + 3*x^7*y + x^6*y^2 - 2*x^5*y^3 + x^4*y^4 + x^3*y^5 - 2*x^2*y^6 + x*y^7 - 3*y^8 - 3*x^5*y^2*z - x^3*y^4*z - 3*x^2*y^5*z - x*y^6*z + 3*x^6*z^2 + 2*x^5*y*z^2 - x^4*y^2*z^2 - 2*x^3*y^3*z^2 - 3*x^5*z^3 + 3*x^4*y*z^3 + x^3*y^2*z^3 - 2*x^2*y^3*z^3 + 2*x*y^4*z^3 - y^5*z^3 - x^3*y*z^4 - 2*x^2*y^2*z^4 + x*y^3*z^4 + 2*y^4*z^4 - x^3*z^5 + x^2*y*z^5 - x*y^2*z^5 - x^2*z^6 - 2*x*y*z^6 + y^2*z^6 - y*z^7 + z^8 + 74717/740880*x^7 - 2*x^6*y + x^5*y^2 + 2*x^4*y^3 - x^3*y^4 - 2*x^2*y^5 - 3*x*y^6 + 3*y^7 - 2*x^6*z - 3*x^5*y*z - 3*x^4*y^2*z + 3*x^3*y^3*z + x^2*y^4*z - 2*x*y^5*z - 2*y^6*z - 3*x^5*z^2 + x^4*y*z^2 + 2*x^3*y^2*z^2 + 2*x^2*y^3*z^2 + 2*x*y^4*z^2 - y^5*z^2 + x^4*z^3 - x^3*y*z^3 - 3*x*y^3*z^3 - 2*x^3*z^4 + 3*x^2*y*z^4 - 3*x*y^2*z^4 + 2*x^2*z^5 + 2*x*y*z^5 + 3*y^2*z^5 + 3*z^7 + 2405/37044*x^6 + x^5*y + 3*x^4*y^2 + 2*x^3*y^3 - 2*x^2*y^4 - x*y^5 + 2*y^6 - 3*x^5*z - 3*x^4*y*z - 2*x^3*y^2*z - 3*x^2*y^3*z + 2*x*y^4*z + y^5*z - x^4*z^2 + x^3*y*z^2 + 2*x^2*y^2*z^2 - x*y^3*z^2 - 3*y^4*z^2 + 2*x^3*z^3 + 2*x^2*y*z^3 - 3*x*y^2*z^3 - 2*y^3*z^3 - 3*x*y*z^4 + 3*y^2*z^4 + 3*x*z^5 + 3*z^6 - 5/42*x^5 + x^4*y + 3*x^3*y^2 - 2*x^2*y^3 - 2*x*y^4 - y^5 - 7/3*x^4*z + x^3*y*z - 2*x^2*y^2*z + 3*x*y^3*z - 3*y^4*z + x^3*z^2 + x^2*y*z^2 + x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 - 3*x*y*z^3 + 2*x*z^4 - 3*y*z^4 - 2*z^5 + 2*x^3*y + 2*x^2*y^2 - 2*x*y^3 + 3*y^4 + 1/2*x^3*z + 3*x^2*y*z + 2*x*y^2*z - 5/2*x^2*z^2 - 2*x*y*z^2 + 2*x*z^3 - 3*y*z^3 + z^4 + 3*x*y^2 - 2*y^3 - 3*x*y*z - y^2*z + 2*y*z^2', '3*x^8 - 3*x^6*y^2 + 3*x^5*y^3 + x^4*y^4 + 2*x^2*y^6 + 3*x*y^7 + x^7*z - x^5*y^2*z - 3*x^4*y^3*z + 3*x^3*y^4*z - x^2*y^5*z - 2*x*y^6*z - 3*y^7*z + 3*x^4*y^2*z^2 - 2*x^3*y^3*z^2 + x^2*y^4*z^2 + 2*x*y^5*z^2 + 3*y^6*z^2 - 2*x^5*z^3 - 2*x^4*y*z^3 - 3*x^3*y^2*z^3 + 2*x^2*y^3*z^3 + x*y^4*z^3 - 2*y^5*z^3 - 2*x^4*z^4 - 3*x^3*y*z^4 + 2*x^2*y^2*z^4 + x*y^3*z^4 + y^4*z^4 - 3*x^3*z^5 + x^2*y*z^5 - 2*x*y^2*z^5 - y^3*z^5 + 2*x^2*z^6 + 2*x*y*z^6 - 3*y^2*z^6 + 3*x*z^7 - 2*y*z^7 + 2*z^8 + 4087/20580*x^7 - 2*x^6*y + 3*x^5*y^2 - 2*x^4*y^3 - 2*x^3*y^4 - x^2*y^5 - 2*x*y^6 + 3*y^7 + 2*x^5*y*z - x^4*y^2*z - x^3*y^3*z - 3*x^2*y^4*z + 3*x*y^5*z + 3*y^6*z - 2*x^5*z^2 + 2*x^4*y*z^2 + x^3*y^2*z^2 + 2*x^2*y^3*z^2 - 2*x*y^4*z^2 + 3*y^5*z^2 - x^3*y*z^3 + 3*x^2*y^2*z^3 - x*y^3*z^3 - 2*y^4*z^3 - x^3*z^4 + x^2*y*z^4 - 3*x*y^2*z^4 + 2*y^3*z^4 + x^2*z^5 + 3*y^2*z^5 + 3*x*z^6 + 3*y*z^6 - 3*z^7 + 21247/185220*x^6 - 3*x^5*y + 3*x^4*y^2 + 3*x^2*y^4 + 3*y^6 - x^5*z - 3*x^4*y*z + 3*x^3*y^2*z - x*y^4*z - 2*y^5*z - 3*x^4*z^2 - 2*x^3*y*z^2 - 3*x^2*y^2*z^2 + 2*x*y^3*z^2 + 2*x^3*z^3 - 2*x^2*y*z^3 - x*y^2*z^3 - 2*y^3*z^3 - 2*x^2*z^4 + x*y*z^4 - 3*y^2*z^4 - 2*x*z^5 + 2*y*z^5 + 3*z^6 - 88/441*x^5 - 2*x^3*y^2 - 7/3*x^4*z + 2*x^3*y*z - 2*x^2*y^2*z + x*y^3*z + y^4*z - 2*x^3*z^2 + x^2*y*z^2 - x*y^2*z^2 + 3*y^3*z^2 - x^2*z^3 - 2*x*y*z^3 - 2*y^2*z^3 - x*z^4 + y*z^4 - z^5 + 3*x^3*y + x*y^3 + 4/3*x^3*z + 2*x^2*y*z + 3*x*y^2*z - 2*y^3*z - 3/2*x^2*z^2 + x*y*z^2 + x*z^3 - 2*y*z^3 + z^4 - x*y^2 - 2*y^3 - 2*x*y*z - 3*y*z^2 - 2*z^3']
  ok   R4 N3, component W4 (P_402 = P_203 = 0, 5P_501^2 = 21P_302): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.2 s
  ok   R4 N3, component W4 (P_402 = P_203 = 0, 5P_501^2 = 21P_302) anchor (A): fixed {'P_402': 0, 'P_203': 0}, solved ['P_302']; member closed with H and the prescribed graded data: True; a triple of I of local length 37 (ds), stabilization (37, 12)
      anchor generators: {'P': 'X^7 + 2*X^5*Z + 20/21*X^3*Z^2 + 2*X*Y*Z - Y^2*Z - Y*Z^2', 'G': 'X^4*Z + X^2*Y^2 - 2*X*Y^2 - Y^3 + X*Y*Z + Y*Z^2 - 2*Z^3', 'H': 'X^2*Z^2 + X*Y^2 - 2*X*Y*Z - Y^2*Z + 2*Y*Z^2 + Z^3'}
      anchor triple: ['x^8 + 3*x^7*y + 2*x^5*y^3 + x^4*y^4 - x^3*y^5 + 3*x^2*y^6 + 3*x*y^7 + 3*x^7*z + x^6*y*z - x^5*y^2*z + 3*x^4*y^3*z - x^3*y^4*z - x^2*y^5*z + y^7*z + 3*x^6*z^2 - 2*x^5*y*z^2 + 2*x^4*y^2*z^2 + 3*x^3*y^3*z^2 + 2*x^2*y^4*z^2 + 2*x*y^5*z^2 + y^6*z^2 + 3*x^5*z^3 + x^4*y*z^3 - 2*x^3*y^2*z^3 - 2*x*y^4*z^3 - 2*y^5*z^3 + 3*x^4*z^4 + 2*x^3*y*z^4 + 2*x*y^3*z^4 - 2*y^4*z^4 - x^3*z^5 + 3*x^2*y*z^5 - 2*x*y^2*z^5 + 2*y^3*z^5 - 2*x^2*z^6 - 3*x*y*z^6 + y^2*z^6 + 2*x*z^7 - 3*y*z^7 + z^8 + 433/8820*x^7 - 3*x^5*y^2 - 3*x^4*y^3 + x^3*y^4 + x*y^6 + 3*y^7 - 2*x^6*z - 2*x^5*y*z + 2*x^4*y^2*z + x^2*y^4*z + y^6*z + 2*x^5*z^2 - 2*x^4*y*z^2 - 2*x^3*y^2*z^2 - x^2*y^3*z^2 - 2*x*y^4*z^2 - y^5*z^2 + 2*x^4*z^3 + 3*x^3*y*z^3 + 2*x^2*y^2*z^3 + x*y^3*z^3 + 2*y^4*z^3 + 3*x^3*z^4 + 3*x^2*y*z^4 - 3*x*y^2*z^4 + y^3*z^4 - 2*x^2*z^5 + 2*x*z^6 + 2*z^7 + 95/3528*x^6 - 2*x^5*y - 2*x^4*y^2 - x^3*y^3 - 2*x^2*y^4 + 3*x*y^5 - 3*y^6 - x^5*z - x^4*y*z - x^3*y^2*z - x^2*y^3*z + y^5*z - 3*x^3*y*z^2 + x^2*y^2*z^2 + 2*x*y^3*z^2 - 2*y^4*z^2 + 2*x^3*z^3 - 3*y^3*z^3 - 3*x^2*z^4 - 3*x*y*z^4 - y^2*z^4 - 3*x*z^5 + y*z^5 + 2*z^6 - 1/126*x^5 + 3*x^4*y - 3*x^3*y^2 + x^2*y^3 + 2*x*y^4 + y^5 - 13/24*x^4*z + 3*x^3*y*z - 2*x^2*y^2*z - 2*x*y^3*z - x^3*z^2 - x*y^2*z^2 + 2*y^3*z^2 + 2*x*y*z^3 + y^2*z^3 + 2*x*z^4 - 3*y*z^4 - 3*z^5 + x^3*y + 2*x^2*y^2 + 2*x*y^3 - 2*y^4 + 1/6*x^3*z + 3*x^2*y*z + 3*x*y^2*z - y^3*z - 1/2*x^2*z^2 + 2*y^2*z^2 + x*z^3 + 2*y*z^3 + z^4 - x*y^2 + y^3 + 3*x*y*z - y^2*z + 2*y*z^2', '-x^8 - 2*x^7*y + 2*x^6*y^2 + x^5*y^3 - 3*x^3*y^5 - x^2*y^6 - x*y^7 - 3*y^8 + 2*x^7*z - 3*x^5*y^2*z - x^3*y^4*z + 2*x^2*y^5*z - 3*x*y^6*z - y^7*z + x^6*z^2 - 3*x^5*y*z^2 + x^4*y^2*z^2 - 2*x^2*y^4*z^2 + y^6*z^2 + 2*x^3*y^2*z^3 + x^2*y^3*z^3 + 3*x*y^4*z^3 - y^5*z^3 + 3*x^4*z^4 + 2*x^3*y*z^4 - x*y^3*z^4 + x^2*y*z^5 + x*y^2*z^5 - y^3*z^5 - 3*x^2*z^6 + 3*x*y*z^6 - y^2*z^6 + 2*x*z^7 + 2*y*z^7 - z^8 + 359/3528*x^7 - x^6*y - 3*x^5*y^2 + 2*x^4*y^3 - x^3*y^4 + x*y^6 + 3*y^7 - 3*x^5*y*z - x^4*y^2*z - x^2*y^4*z + x*y^5*z - y^6*z + 2*x^5*z^2 - x^4*y*z^2 + 2*x^3*y^2*z^2 + 3*x^2*y^3*z^2 + 3*x*y^4*z^2 + y^5*z^2 + 2*x^4*z^3 - 2*y^4*z^3 - 2*x^3*z^4 - x^2*y*z^4 - 2*x*y^2*z^4 - y^3*z^4 - x^2*z^5 - 3*x*y*z^5 - 3*y^2*z^5 - 3*x*z^6 - 2*z^7 + 167/3528*x^6 - 3*x^4*y^2 - x^3*y^3 + 3*x^2*y^4 - 2*y^6 - 2*x^5*z + x^4*y*z + x^2*y^3*z - y^5*z - x^4*z^2 - x^3*y*z^2 - 2*x^2*y^2*z^2 - 3*x*y^3*z^2 - 2*y^4*z^2 - 3*x^3*z^3 - 3*x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 3*x*y*z^4 - x*z^5 - 3*y*z^5 + 2*z^6 + 1/63*x^5 - 3*x^4*y + x^3*y^2 - 2*x^2*y^3 + 3*x*y^4 - 2*y^5 - 25/24*x^4*z + 3*x^3*y*z - 2*x^2*y^2*z + 3*x*y^3*z + 2*y^4*z - 2*x^3*z^2 + 3*x^2*y*z^2 + 3*x*y^2*z^2 - y^3*z^2 + 3*x^2*z^3 - 3*x*y*z^3 + 3*y^2*z^3 + 3*x*z^4 - 2*y*z^4 + z^5 - 2*x^3*y + 2*x^2*y^2 + x*y^3 - y^4 - 1/3*x^3*z - 3*x^2*y*z + x*y^2*z + 2*y^3*z + x^2*z^2 - x*y*z^2 + 2*y^2*z^2 + 2*x*z^3 + y*z^3 + 2*z^4 + 2*x*y^2 + y^3 + x*y*z + 3*y^2*z + 3*y*z^2 - 2*z^3', '3*x^8 + x^6*y^2 - 3*x^5*y^3 + 2*x^3*y^5 + 2*x^2*y^6 - 2*x*y^7 - 2*y^8 - 2*x^7*z + 2*x^5*y^2*z + 3*x^4*y^3*z - x^2*y^5*z - y^7*z - x^6*z^2 + 2*x^5*y*z^2 + 3*x^4*y^2*z^2 + 3*x^3*y^3*z^2 - x^2*y^4*z^2 + 2*x*y^5*z^2 + 2*y^6*z^2 + 3*x^5*z^3 - x^4*y*z^3 - 2*x^3*y^2*z^3 + 3*x^2*y^3*z^3 - x*y^4*z^3 - 2*y^5*z^3 + x^4*z^4 + x^3*y*z^4 + x^2*y^2*z^4 + 2*x*y^3*z^4 - 2*x^3*z^5 - 2*x*y^2*z^5 + 3*y^3*z^5 - 3*x^2*z^6 - 2*x*y*z^6 + y^2*z^6 - x*z^7 - y*z^7 - 3*z^8 + 193/4410*x^7 + x^6*y + 3*x^5*y^2 + 2*x^4*y^3 - 3*x^3*y^4 + 3*x^2*y^5 + 2*x*y^6 + y^7 - x^4*y^2*z - x*y^5*z + 2*y^6*z - 2*x^4*y*z^2 + 2*x^3*y^2*z^2 - 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 + 2*x^4*z^3 - 2*x^3*y*z^3 - x^2*y^2*z^3 - 3*x*y^3*z^3 + y^4*z^3 + 2*x^3*z^4 + x^2*y*z^4 - 2*x*y^2*z^4 - y^3*z^4 - 2*x^2*z^5 - 3*x*y*z^5 - 3*x*z^6 + 3*z^7 + 5/1764*x^6 + x^5*y + 3*x^4*y^2 + x^3*y^3 + 2*x^2*y^4 + 3*x*y^5 - x^5*z + x^4*y*z - x^3*y^2*z - 3*x^2*y^3*z - 2*x*y^4*z - y^5*z + 3*x^3*y*z^2 - 3*x*y^3*z^2 + 3*x^3*z^3 - 2*x^2*y*z^3 + 2*y^3*z^3 - x^2*z^4 - 2*x*y*z^4 - 3*y^2*z^4 - x*z^5 - y*z^5 + 3*z^6 - 1/126*x^5 + x^4*y + 2*x^3*y^2 + 3*x^2*y^3 - 2*x*y^4 - 2*y^5 - 1/12*x^4*z + 3*x^3*y*z + 3*x*y^3*z - y^4*z + x^3*z^2 + x^2*y*z^2 + x*y^2*z^2 + y^3*z^2 + 2*x^2*z^3 + 2*x*y*z^3 - 3*y^2*z^3 - 2*y*z^4 - 3*x^3*y - 3*x^2*y^2 - y^4 + 1/6*x^3*z - x^2*y*z + 3*x*y^2*z + 2*y^3*z + 1/2*x^2*z^2 - 3*x*y*z^2 - 3*y^2*z^2 - 3*x*z^3 + z^4 - x*y^2 + y^3 + 2*x*y*z - 3*y^2*z + y*z^2 - z^3']
--- R4 N7
  ok   R4 N7: the graded normal form is closed with H = [1, 3, 6, 4, 3, 2, 1, 1]
  ok   R4 N7: one lift per pivot of D_4..D_7
    134 closedness conditions in 88 tail coefficients; 56 eliminated with constant pivots; residual equations:
       (-2) * G_220 * P_402
       (-1/42) * (-10*P_501^2*G_220 - 4*P_501*P_402*G_211 - 4*P_510*P_402*G_202 - 14*P_402*G_211*G_202 + 8*P_501*P_402*H_120 + 56*P_402*G_202*H_120 - 4*P_510*P_402*H_111 + 14*P_402*G_211*H_111 + 6*P_510*P_203 + 42*P_302*G_220 + 21*P_203*G_211 + 84*P_402*G_120 - 84*P_203*H_120)
       (-2/63) * (6*P_501*P_402*H_030 + 42*P_402*G_202*H_030 - 2*P_510*P_402*H_021 + 7*P_402*G_211*H_021 + 63*P_402*G_030 - 63*P_203*H_030)
       (-1/21) * (-P_510*P_402*G_211 + 4*P_501*P_402*H_021 + 28*P_402*G_202*H_021 - 4*P_510*P_402*H_012 + 14*P_402*G_211*H_012 + 42*P_402*G_021 - 42*P_203*H_021)
       (-1/42) * (2*P_501*P_402*G_211 - 6*P_501*P_402*G_103 - 42*P_402*G_202*G_103 + 8*P_501*P_402*H_012 + 56*P_402*G_202*H_012 - 24*P_510*P_402*H_003 + 84*P_402*G_211*H_003 - 6*P_510*P_203 + 63*P_203*G_103 + 84*P_402*G_012 - 84*P_203*H_012)
       (-4/7) * G_220 * P_402
  ok   R4 N7: on {P_402 != 0} the residual equations are triangular in ['G_220', 'G_120', 'G_030', 'G_021', 'G_012'] with diagonal (const)*P_402: a graph (irreducible)
  ok   R4 N7: on {P_402 = 0}: on {P_203 != 0} triangular in ['H_120', 'H_030', 'H_021', 'H_012'] with diagonal (const)*P_203; on {P_203 = 0}: multiples of 5*P_501^2*G_220 - 21*P_302*G_220: ['(-1/21) * G_220 * (-5*P_501^2 + 21*P_302)']
  ok   R4 N7, component W1 (P_402 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N7, component W1 (P_402 != 0) anchor (A): fixed {'P_402': 1}, solved ['G_220', 'G_120', 'G_030', 'G_021', 'G_012']; member closed with H and the prescribed graded data: True; a triple of I of local length 37 (ds), stabilization (37, 11)
      anchor generators: {'P': 'X^7 + X^4*Z^2 + X*Y*Z^2 - Z^4 + X*Y^2 - 2*Y^3 - 2*X*Y*Z + Y^2*Z + 2*Y*Z^2 - Z^3', 'G': 'X^4*Z + X*Y*Z - 2*Z^3', 'H': 'X^3*Y - 3/2*X^2*Z^2 + 2*X*Y^2 - 2*X*Y*Z - 2*Y^2*Z + 2*Y*Z^2 + 2*Z^3'}
      anchor triple: ['-3*x^8 - 2*x^7*y + x^6*y^2 - 3*x^5*y^3 - 3*x^4*y^4 + x^3*y^5 - 2*x^2*y^6 - 3*x*y^7 + 3*x^7*z - x^6*y*z + x^5*y^2*z - 2*x^4*y^3*z + 3*x^3*y^4*z + 2*x^2*y^5*z + y^7*z - 2*x^6*z^2 + x^4*y^2*z^2 + x^3*y^3*z^2 - x^2*y^4*z^2 + x*y^5*z^2 + 2*y^6*z^2 - 2*x^5*z^3 + 2*x^4*y*z^3 + 3*x^2*y^3*z^3 - 3*x*y^4*z^3 - 3*y^5*z^3 + 2*x^4*z^4 - 2*x^3*y*z^4 + 2*x*y^3*z^4 - 2*x^3*z^5 - 2*x^2*y*z^5 + x*y^2*z^5 - 2*y^3*z^5 + x^2*z^6 + 3*x*y*z^6 - 2*y^2*z^6 - 3*x*z^7 + y*z^7 + z^8 + 41/2520*x^7 + 2*x^6*y - 3*x^5*y^2 + x^4*y^3 + 2*x^3*y^4 + x^2*y^5 + 3*x*y^6 + 3*y^7 - x^4*y^2*z + 2*x^3*y^3*z - 2*x^2*y^4*z - x*y^5*z - 3*x^5*z^2 + x^4*y*z^2 + 3*x^3*y^2*z^2 + 3*x^2*y^3*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 + 2*x^3*y*z^3 + x^2*y^2*z^3 - 3*y^4*z^3 - 3*x^3*z^4 + x*y^2*z^4 + y^3*z^4 + x^2*z^5 - x*y*z^5 - 3*y^2*z^5 + 3*x*z^6 + z^7 + 3*x^5*y - 2*x^4*y^2 - x^2*y^4 + 3*x*y^5 - y^6 + 3*x^5*z + 2*x^4*y*z - 3*x^2*y^3*z + 3*x*y^4*z - 2*x^4*z^2 + 2*x^3*y*z^2 - 2*x^2*y^2*z^2 + 2*x*y^3*z^2 - y^4*z^2 + 3*x^2*y*z^3 - x*y^2*z^3 + 3*y^3*z^3 + 3*x^2*z^4 - x*y*z^4 - 3*y^2*z^4 + x*z^5 + 1/105*x^5 - 2*x^4*y + 3*x^3*y^2 + x^2*y^3 + 2*x*y^4 - y^5 + 23/24*x^4*z - 2*x^3*y*z + x^2*y^2*z + x*y^3*z + 3*y^4*z - x^2*y*z^2 + x*y^2*z^2 - 3*y^3*z^2 - x^2*z^3 + 2*x*z^4 + 2*y*z^4 + 3*z^5 - 16/3*x^3*y + x^2*y^2 + 2*x*y^3 + y^4 + x^2*y*z + 3*x*y^2*z - 2*y^3*z - x^2*z^2 - x*y*z^2 - 3*y^2*z^2 - 2*x*z^3 - 2*y*z^3 - 2*z^4 + 2*x*y^2 + 2*y^3 + x*y*z + y^2*z + 2*z^3', '-x^8 + x^6*y^2 + 2*x^5*y^3 - 3*x^4*y^4 - x^3*y^5 - 3*x*y^7 + 3*y^8 + 2*x^7*z + x^6*y*z + x^4*y^3*z + x^3*y^4*z + x^2*y^5*z - 2*x*y^6*z - 2*y^7*z - x^6*z^2 - 3*x^5*y*z^2 - 2*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + x^2*y^4*z^2 - 3*x*y^5*z^2 + 2*y^6*z^2 - 3*x^5*z^3 - 2*x^4*y*z^3 + x^3*y^2*z^3 + x^2*y^3*z^3 + 3*x*y^4*z^3 - y^5*z^3 + 3*x^4*z^4 + 2*x^3*y*z^4 - 2*x^2*y^2*z^4 + 3*x*y^3*z^4 + y^4*z^4 - x^2*y*z^5 + y^3*z^5 + 3*x^2*z^6 + 2*x*y*z^6 - y^2*z^6 - 2*z^8 + 13/840*x^7 + 2*x^6*y + x^5*y^2 + x^4*y^3 + x^3*y^4 + x*y^6 - 3*y^7 + 3*x^6*z - 2*x^5*y*z + x^4*y^2*z - 3*x^3*y^3*z + 3*x^2*y^4*z + 2*x*y^5*z + 3*y^6*z - x^5*z^2 + x^4*y*z^2 - 2*x^3*y^2*z^2 - 3*x^2*y^3*z^2 - x*y^4*z^2 + 3*y^5*z^2 - 2*x^4*z^3 + x^3*y*z^3 + x^2*y^2*z^3 - x*y^3*z^3 + 2*y^4*z^3 + 3*x^3*z^4 + 3*x^2*y*z^4 - 3*x^2*z^5 + 2*x*y*z^5 - x*z^6 + y*z^6 - 2*z^7 + 1/840*x^6 - 3*x^5*y - x^4*y^2 + 2*x^3*y^3 - x*y^5 - 2*y^6 - 3*x^5*z + x^4*y*z + 2*x^3*y^2*z - 3*x^2*y^3*z - x*y^4*z - 3*y^5*z - 3*x^4*z^2 + x^3*y*z^2 + 3*x^2*y^2*z^2 - x*y^3*z^2 - y^4*z^2 + 3*x^2*y*z^3 - 2*x*y^2*z^3 + y^3*z^3 + 2*x^2*z^4 - x*y*z^4 + 2*y^2*z^4 - 3*x*z^5 - z^6 + 2/105*x^5 + x^4*y + 3*x^3*y^2 - 3*x^2*y^3 - x*y^4 - y^5 + 23/24*x^4*z + 3*x^3*y*z + 3*x^2*y^2*z + 2*x*y^3*z - 3*y^4*z + 2*x^2*y*z^2 - 3*x*y^2*z^2 + y^3*z^2 - x^2*z^3 + x*y*z^3 + 2*y^2*z^3 - x*z^4 + 2*y*z^4 + 3*z^5 - 5*x^3*y - 2*x^2*y^2 - 2*x*y^3 + 2*y^4 - 3*x^2*y*z - x*y^2*z + 2*y^3*z - 2*x^2*z^2 + 3*y^2*z^2 + x*z^3 + y*z^3 - 3*z^4 - 2*y^3 + x*y*z - 2*y^2*z - 3*y*z^2 + 2*z^3', '-x^7*y - 2*x^6*y^2 - 3*x^5*y^3 - 3*x^4*y^4 - 3*x^3*y^5 - 2*x^2*y^6 + 3*x*y^7 - 3*y^8 + x^7*z - 2*x^6*y*z + 2*x^5*y^2*z - 2*x^4*y^3*z - x^3*y^4*z + x^2*y^5*z - x*y^6*z + y^7*z - x^6*z^2 + 2*x^5*y*z^2 + 3*x^4*y^2*z^2 + x^3*y^3*z^2 + 2*x^2*y^4*z^2 - 3*x*y^5*z^2 - 2*y^6*z^2 + 2*x^5*z^3 - 3*x^4*y*z^3 - 2*x^2*y^3*z^3 + 2*y^5*z^3 + x^4*z^4 + x^3*y*z^4 - x^2*y^2*z^4 - 2*x*y^3*z^4 - 3*y^4*z^4 + 2*x^3*z^5 - 3*x^2*y*z^5 + x*y^2*z^5 + 3*y^3*z^5 + 2*x^2*z^6 + 3*x*y*z^6 + 2*y^2*z^6 + 3*z^8 - 11/1260*x^7 + x^6*y - 3*x^5*y^2 + 2*x^4*y^3 + 2*x^3*y^4 + 3*x^2*y^5 + 2*x*y^6 - y^7 + x^6*z - 2*x^5*y*z + 3*x^4*y^2*z + 2*x^3*y^3*z + 3*x^2*y^4*z - 3*x*y^5*z + y^6*z - 2*x^5*z^2 - 2*x^4*y*z^2 + 2*x^3*y^2*z^2 + 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - y^5*z^2 - 3*x^4*z^3 + x^3*y*z^3 - 2*x^2*y^2*z^3 + x*y^3*z^3 - x^3*z^4 - 3*x^2*y*z^4 + 2*x*y^2*z^4 - y^3*z^4 - 2*x^2*z^5 + 2*x*y*z^5 + 2*y^2*z^5 - x*z^6 - y*z^6 + 3*z^7 - 13/1260*x^6 + 2*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 3*x^2*y^4 - x*y^5 - 3*y^6 + x^5*z - 3*x^4*y*z - 3*x^3*y^2*z + 2*x^2*y^3*z + y^5*z + 2*x^4*z^2 + x^3*y*z^2 - 3*x^2*y^2*z^2 - x^3*z^3 - x^2*y*z^3 + 2*x*y^2*z^3 - 2*y^3*z^3 + 3*x^2*z^4 + 2*x*y*z^4 - x*z^5 + 3*y*z^5 + 3*x^4*y - 2*x^3*y^2 - 3*x^2*y^3 - 2*x*y^4 + 3*y^5 - 5/8*x^4*z - 2*x^3*y*z - 2*x^2*y^2*z + 3*x*y^3*z + x^3*z^2 - 3*x^2*y*z^2 - y^3*z^2 - 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 - 2*y*z^4 + x^3*y + 2*x^2*y^2 + x*y^3 - 2*y^4 + 2*x*y^2*z - 3*x*y*z^2 - 3*y^2*z^2 - 2*x*z^3 + 2*y*z^3 + 2*z^4 - x*y^2 + 3*x*y*z - 2*y^2*z + 2*y*z^2 - z^3']
  ok   R4 N7, component W2 (P_402 = 0, P_203 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N7, component W2 (P_402 = 0, P_203 != 0) anchor (A): fixed {'P_402': 0, 'P_203': 1}, solved ['H_120', 'H_030', 'H_021', 'H_012']; member closed with H and the prescribed graded data: True; a triple of I of local length 39 (ds), stabilization (39, 11)
      anchor generators: {'P': 'X^7 - 2*X^3*Y*Z + X^2*Z^3 + X*Y*Z^2 - Z^4 + 2*X*Y^2 + 2*Y^3 + X*Y*Z + Y^2*Z - 2*Y*Z^2 + Z^3', 'G': 'X^4*Z - 2*X*Y^2 + Y^3 + 2*X*Y*Z + Y^2*Z + Y*Z^2 - Z^3', 'H': 'X^3*Y - 3/2*X^2*Z^2 - X*Y*Z + 2*Z^3'}
      anchor triple: ['2*x^8 + x^7*y - 2*x^5*y^3 - x^4*y^4 - 3*x^2*y^6 - 2*x*y^7 + 2*y^8 - 3*x^7*z - 3*x^6*y*z - 2*x^4*y^3*z + x^3*y^4*z - x^2*y^5*z + 3*x*y^6*z + 3*y^7*z - 2*x^6*z^2 - 3*x^5*y*z^2 + 2*x^4*y^2*z^2 - 3*x^3*y^3*z^2 - 2*x^2*y^4*z^2 - x*y^5*z^2 + 3*x^5*z^3 - 3*x^4*y*z^3 - 2*x^3*y^2*z^3 - 3*x^2*y^3*z^3 - 3*x*y^4*z^3 + 3*y^5*z^3 - x^3*y*z^4 - 2*x*y^3*z^4 + 3*y^4*z^4 - 2*x^3*z^5 + 3*x^2*y*z^5 + 3*y^3*z^5 + 2*x*y*z^6 - y^2*z^6 + 2*x*z^7 - 3*y*z^7 - 1/48*x^7 - x^6*y - 2*x^5*y^2 + 2*x^4*y^3 + x^2*y^5 - x*y^6 + y^7 + x^5*y*z - 2*x^4*y^2*z + x^3*y^3*z - x^2*y^4*z - 2*x*y^5*z - 2*y^6*z + 2*x^5*z^2 - 2*x^4*y*z^2 - x^2*y^3*z^2 + 2*x*y^4*z^2 - y^5*z^2 - 3*x^4*z^3 + 3*x^3*y*z^3 - 2*x^2*y^2*z^3 + 2*x*y^3*z^3 - 2*x^3*z^4 - x^2*y*z^4 - 2*y^3*z^4 + 3*x^2*z^5 + 2*x*y*z^5 + y^2*z^5 + 3*x*z^6 - 2*y*z^6 + 2*z^7 + 1/252*x^6 + x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 3*x^2*y^4 + x*y^5 + y^6 - x^5*z + 3*x^4*y*z + x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + x^3*y*z^2 + 2*y^4*z^2 - x^2*y*z^3 - 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - 3*y^2*z^4 + x*z^5 - 2*y*z^5 - 1/105*x^5 + 3*x^4*y - x^3*y^2 - x^2*y^3 - 3*x*y^4 - 3*y^5 + x^4*z - 2*x^3*y*z - x^2*y^2*z - x*y^3*z - 2*y^4*z + 3*x^3*z^2 + 2*x^2*y*z^2 + 3*y^3*z^2 + 3*x^2*z^3 - 2*x*y*z^3 - y^2*z^3 + 2*x*z^4 + 3*y*z^4 - 3*z^5 - 3/2*x^3*y - 3*x^2*y^2 + 2*x*y^3 + 3*y^4 + x^2*y*z - x*y^2*z - 3*y^3*z + x^2*z^2 - 3*x*y*z^2 + 3*y^2*z^2 - x*z^3 + 3*y*z^3 - 2*z^4 + 3*x*y^2 - 3*x*y*z - 2*y^2*z + 2*y*z^2 + z^3', '-3*x^8 + 2*x^7*y + x^6*y^2 + x^5*y^3 - x^4*y^4 - 3*x^2*y^6 - 2*x*y^7 + 3*x^7*z - 2*x^4*y^3*z - 3*x^3*y^4*z + 3*x*y^6*z + 3*y^7*z - 2*x^6*z^2 - x^5*y*z^2 + x^4*y^2*z^2 + 3*x^3*y^3*z^2 - x^2*y^4*z^2 + 3*x*y^5*z^2 - 2*x^5*z^3 - 2*x^4*y*z^3 - x^3*y^2*z^3 - x^2*y^3*z^3 + 3*x*y^4*z^3 - y^5*z^3 - x^4*z^4 - 3*x^3*y*z^4 + 3*x^2*y^2*z^4 + x*y^3*z^4 + 3*y^4*z^4 + x^3*z^5 - 3*x^2*y*z^5 - 2*x*y^2*z^5 - y^3*z^5 - 3*x^2*z^6 - y^2*z^6 + x*z^7 + 2*y*z^7 + 3*z^8 + 13/2520*x^7 + 3*x^6*y - x^5*y^2 - x^4*y^3 - 2*x^3*y^4 - 3*x^2*y^5 - 2*x*y^6 + 2*y^7 - x^4*y^2*z + x^3*y^3*z - x^2*y^4*z - x*y^5*z + y^6*z - x^5*z^2 + 2*x^3*y^2*z^2 - 2*x^2*y^3*z^2 + y^5*z^2 - x^4*z^3 + x^3*y*z^3 - 3*x^2*y^2*z^3 - x*y^3*z^3 - y^4*z^3 + 3*x*y^2*z^4 - y^3*z^4 - 2*x^2*z^5 - 3*x*y*z^5 - 3*y^2*z^5 - x*z^6 + 3*y*z^6 + 3*z^7 + 17/2520*x^6 - x^5*y + 3*x^4*y^2 + x^3*y^3 + 3*x^2*y^4 - 2*x*y^5 + y^6 + x^4*y*z + 3*x^3*y^2*z + 3*x^2*y^3*z + x*y^4*z + 3*y^5*z + 3*x^4*z^2 + 3*x^3*y*z^2 + x^2*y^2*z^2 - 3*x*y^3*z^2 - 2*y^4*z^2 - x^3*z^3 + 2*x^2*y*z^3 + 3*x*y^2*z^3 - y^3*z^3 + 2*x^2*z^4 + 3*y^2*z^4 + x*z^5 - 2*y*z^5 - z^6 - 1/140*x^5 - x^4*y + x^3*y^2 - 2*x*y^4 + 2*y^5 + 4/3*x^4*z - x^3*y*z + 3*x^2*y^2*z - 3*x*y^3*z - 3*y^4*z + 3*x^2*y*z^2 - 3*x*y^2*z^2 + y^3*z^2 - 2*x*y*z^3 - 3*y^2*z^3 + z^5 - 4/3*x^3*y + x^2*y^2 + 2*x*y^3 - 2*y^4 + 3*x^2*y*z - 2*x*y^2*z + 3*y^3*z + x^2*z^2 - 2*x*y*z^2 + 2*y*z^3 + x*y^2 - 3*y^3 - 2*x*y*z - y^2*z + y*z^2 + z^3', 'x^8 - 2*x^7*y + 2*x^6*y^2 + x^5*y^3 - 2*x^4*y^4 - 3*x^3*y^5 - 3*x^2*y^6 - 2*x*y^7 - y^8 + x^7*z - 3*x^6*y*z + 2*x^5*y^2*z - 2*x^4*y^3*z + 3*x^3*y^4*z + 2*x^2*y^5*z - 3*x*y^6*z - 3*y^7*z - x^6*z^2 + 3*x^5*y*z^2 - 2*x^4*y^2*z^2 + 2*x^3*y^3*z^2 - x^2*y^4*z^2 - 3*x*y^5*z^2 - 2*y^6*z^2 - 2*x^5*z^3 - x^4*y*z^3 + 2*x^3*y^2*z^3 - 2*x^2*y^3*z^3 - 3*x*y^4*z^3 + 3*y^5*z^3 + 3*x^4*z^4 - 3*x^3*y*z^4 - 3*x^2*y^2*z^4 + 3*x*y^3*z^4 - 3*y^4*z^4 + 2*x^3*z^5 + 2*x*y^2*z^5 - 3*y^3*z^5 + 2*x^2*z^6 + x*y*z^6 - y^2*z^6 - x*z^7 - 2*y*z^7 + z^8 - 2/315*x^7 + 3*x^6*y + 3*x^5*y^2 + x^2*y^5 + 2*x*y^6 - 3*y^7 + x^6*z + 3*x^5*y*z + 2*x^4*y^2*z - x^3*y^3*z + x^2*y^4*z - 2*x*y^5*z - y^6*z + 3*x^5*z^2 + x^4*y*z^2 - x^3*y^2*z^2 - 2*x^2*y^3*z^2 - 2*x*y^4*z^2 + 3*y^5*z^2 + 3*x^4*z^3 - 2*x^3*y*z^3 + 3*x^2*y^2*z^3 - x*y^3*z^3 - y^4*z^3 - x^3*z^4 + 3*x^2*y*z^4 - y^3*z^4 - 3*x^2*z^5 - x*y*z^5 - y^2*z^5 + 3*x*z^6 + 2*z^7 - 1/630*x^6 - x^5*y - x^4*y^2 + 2*x^3*y^3 + x*y^5 + y^6 - 3*x^5*z + 3*x^4*y*z + 3*x^3*y^2*z + x^2*y^3*z + 3*x*y^4*z - 2*y^5*z - 2*x^4*z^2 - x^3*y*z^2 + 2*x^2*y^2*z^2 - x*y^3*z^2 + y^4*z^2 - x^3*z^3 + x*y^2*z^3 + 2*x^2*z^4 - 2*y^2*z^4 - 1/140*x^5 + 2*x^4*y - 2*x^3*y^2 - x^2*y^3 - 2*x*y^4 + 2*y^5 - 1/4*x^4*z + 2*x^3*y*z + 2*x^2*y^2*z - 2*x*y^3*z - y^4*z + 2*x^3*z^2 + 3*x*y^2*z^2 + y^3*z^2 + 3*x^2*z^3 - 3*x*y*z^3 - y^2*z^3 + 3*x*z^4 + y*z^4 + z^5 - 5*x^3*y - 3*x^2*y^2 - 3*x*y^3 + 3*y^4 + x^2*y*z - x*y^2*z - 3*y^3*z + x^2*z^2 - 2*x*y*z^2 - 2*y^2*z^2 + 2*x*z^3 - 2*y*z^3 + z^4 - 3*x*y^2 + 3*y^3 - y^2*z - 2*y*z^2 + 3*z^3']
  ok   R4 N7, component W3 (P_402 = P_203 = 0, G_220 = 0): a point of the component (residual equations vanish)
      timing: member+triples 0.5 s, stabilization 0.2 s
  ok   R4 N7, component W3 (P_402 = P_203 = 0, G_220 = 0) anchor (A): fixed {'P_402': 0, 'P_203': 0, 'G_220': 0}, solved []; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 12)
      anchor generators: {'P': 'X^7 - X^5*Y + X^5*Z + 2*X^4*Y + 5/21*X^3*Y^2 - 10/21*X^3*Y*Z + 2*X^3*Z^2 - X^2*Y^2 - 5/441*X*Y^3 + 2*X^2*Y*Z + 5/147*X*Y^2*Z - 269/294*X*Y*Z^2 + X*Z^3 - 37/168*Z^4 + X*Y^2 - Y^3 - X*Y*Z + Y^2*Z - 2*Y*Z^2', 'G': 'X^4*Z - X^2*Y*Z + X^2*Z^2 - X*Z^3 + 2*X*Y^2 - 2*Y^3 - X*Y*Z - 2*Y^2*Z + Y*Z^2 + Z^3', 'H': 'X^3*Y - 3/2*X^2*Z^2 - X*Y^2 + 2*Y^3 - 2*X*Y*Z - 2*Y^2*Z + Z^3'}
      anchor triple: ['-x^8 - 2*x^7*y + 3*x^6*y^2 + 3*x^5*y^3 + x^4*y^4 - 2*x^3*y^5 - 2*x^2*y^6 - 3*x*y^7 + y^8 - 3*x^6*y*z - x^5*y^2*z - 3*x^4*y^3*z + x^3*y^4*z + 3*x^2*y^5*z - 3*x^6*z^2 - x^5*y*z^2 - x^4*y^2*z^2 + 3*x^3*y^3*z^2 + 2*x*y^5*z^2 + x^5*z^3 - 3*x^4*y*z^3 - 3*x^3*y^2*z^3 + x^2*y^3*z^3 + 2*x*y^4*z^3 + x^4*z^4 - 2*x^3*y*z^4 - x^2*y^2*z^4 - 2*x*y^3*z^4 + y^4*z^4 - x^3*z^5 + 2*x*y^2*z^5 - 3*y^3*z^5 + y^2*z^6 + 2*x*z^7 - 2*y*z^7 + 2*z^8 + 11033/148176*x^7 + 2*x^6*y + x^5*y^2 - x^4*y^3 + 2*x^3*y^4 + 3*x^2*y^5 + 2*x*y^6 - 3*x^6*z + 2*x^5*y*z + 3*x^4*y^2*z - 3*x^3*y^3*z - 3*x^2*y^4*z + 2*x*y^5*z + 3*y^6*z + x^5*z^2 + x^4*y*z^2 + x^3*y^2*z^2 - x^2*y^3*z^2 + 2*x*y^4*z^2 + 3*y^5*z^2 - 2*x^2*y^2*z^3 - 3*x*y^3*z^3 + 2*y^4*z^3 + x^3*z^4 + 2*x^2*y*z^4 + 3*x*y^2*z^4 + 2*y^3*z^4 + 3*x^2*z^5 + 2*x*y*z^5 + 3*y^2*z^5 + 3*x*z^6 - 3*y*z^6 - z^7 + 2671/246960*x^6 + 3*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - x^2*y^4 - y^6 + x^5*z + x^3*y^2*z - 2*x*y^4*z + y^5*z + 3*x^4*z^2 - 3*x^3*y*z^2 + x^2*y^2*z^2 + x*y^3*z^2 - 2*y^4*z^2 + x^2*y*z^3 + 3*x*y^2*z^3 + 2*y^3*z^3 + 3*x*y*z^4 + 3*y^2*z^4 + x*z^5 - z^6 - 89/882*x^5 - 2*x^4*y + 2*x^3*y^2 - 2*x^2*y^3 + 3*x*y^4 - 2*y^5 - 2/3*x^4*z + x^3*y*z + 2*x^2*y^2*z + x*y^3*z - y^4*z - x^3*z^2 - 2*x^2*y*z^2 - 3*x*y^2*z^2 + 2*y^3*z^2 + x^2*z^3 + x*y*z^3 - 3*y^2*z^3 - 2*x*z^4 + y*z^4 - z^5 - 11/3*x^3*y - 2*x^2*y^2 + 3*x*y^3 + y^4 + 2/3*x^3*z - 2*x^2*y*z + 2*x*y^2*z - 3*y^3*z - 3*x^2*z^2 + 3*x*y*z^2 - 2*x*z^3 - 2*y*z^3 + 2*z^4 - 2*x*y^2 - y^3 + 2*x*y*z - y^2*z - 3*y*z^2 + 2*z^3', '3*x^8 - x^7*y + 2*x^6*y^2 - 2*x^4*y^4 + 3*x^3*y^5 - 2*x^2*y^6 - x*y^7 - 2*y^8 - x^7*z + 3*x^4*y^3*z + 3*x^3*y^4*z - 2*x^2*y^5*z - 2*x*y^6*z - y^7*z - 2*x^6*z^2 - 2*x^5*y*z^2 - 2*x^4*y^2*z^2 - x^3*y^3*z^2 - 2*x^2*y^4*z^2 - 3*x*y^5*z^2 + 2*y^6*z^2 - x^5*z^3 + 2*x^4*y*z^3 + 3*x^3*y^2*z^3 + 3*x*y^4*z^3 + 3*y^5*z^3 - 2*x^4*z^4 + x^3*y*z^4 + 3*x^2*y^2*z^4 + 3*x*y^3*z^4 + 2*y^4*z^4 + 2*x^3*z^5 - 3*x^2*y*z^5 + 3*x*y^2*z^5 - y^3*z^5 - 3*x*y*z^6 - 2*y^2*z^6 - x*z^7 - z^8 + 29153/370440*x^7 - 2*x^6*y - x^5*y^2 + 2*x^3*y^4 - 3*x^2*y^5 + 3*x*y^6 + y^7 - 3*x^6*z - 2*x^5*y*z + 3*x^4*y^2*z - 3*x*y^5*z + 3*y^6*z - x^5*z^2 + 3*x^4*y*z^2 - 2*x^3*y^2*z^2 - x^2*y^3*z^2 - x*y^4*z^2 - y^5*z^2 - x^3*y*z^3 - 2*x^2*y^2*z^3 + 3*x*y^3*z^3 - 2*y^4*z^3 + 2*x^3*z^4 + x^2*y*z^4 - y^3*z^4 + 3*x^2*z^5 - 2*x*y*z^5 - 2*y^2*z^5 + 3*x*z^6 + 2*y*z^6 + z^7 - 2881/52920*x^6 - x^4*y^2 + 3*x^3*y^3 + x^2*y^4 - x*y^5 - 2*y^6 - 2*x^5*z + x^4*y*z - 2*x^3*y^2*z - 2*x^2*y^3*z - 2*x*y^4*z - 2*x^3*y*z^2 + y^4*z^2 - x^3*z^3 + x^2*y*z^3 - 3*x*y^2*z^3 + 2*y^3*z^3 - 2*x^2*z^4 + x*y*z^4 + 2*y*z^5 + 115/1176*x^5 - 3*x^4*y - 2*x^2*y^3 - 2*x*y^4 - 49/24*x^4*z + x^3*y*z + 2*x^2*y^2*z - y^4*z - x^3*z^2 + 2*x^2*y*z^2 + 2*y^3*z^2 + 2*x^2*z^3 + 3*x*y*z^3 - 2*y*z^4 + 13/3*x^3*y - x^2*y^2 + 2*x*y^3 - 2*y^4 + 1/4*x^3*z - 2*x*y^2*z - 2*x^2*z^2 - 3*x*y*z^2 - y^2*z^2 - x*z^3 - 3*y*z^3 + 2*x*y^2 - 3*y^3 + 3*x*y*z - 2*y^2*z + y*z^2', '2*x^8 - x^7*y + 2*x^6*y^2 - x^5*y^3 - 2*x^4*y^4 + x^3*y^5 + 2*x^2*y^6 - x*y^7 + 2*x^7*z - 3*x^6*y*z - x^4*y^3*z + 2*x*y^6*z + y^7*z + x^6*z^2 - x^5*y*z^2 - 2*x^4*y^2*z^2 - 3*x^3*y^3*z^2 + x^2*y^4*z^2 - 3*x*y^5*z^2 + y^6*z^2 + x^2*y^3*z^3 - 2*x*y^4*z^3 - y^5*z^3 - 2*x^3*y*z^4 - 2*x^2*y^2*z^4 - 3*x*y^3*z^4 + y^4*z^4 + x^3*z^5 + x^2*y*z^5 - 3*x*y^2*z^5 - y^3*z^5 + 2*x^2*z^6 + 2*x*y*z^6 + 3*y^2*z^6 + x*z^7 + 3*y*z^7 - 2*z^8 + 3971/74088*x^7 + 2*x^6*y + 2*x^4*y^3 + x^3*y^4 - x^2*y^5 - x*y^6 + 2*y^7 + 3*x^6*z - 3*x^5*y*z - 3*x^4*y^2*z - 2*x^3*y^3*z + 2*x^2*y^4*z + 3*x*y^5*z - y^6*z + x^5*z^2 - 3*x^4*y*z^2 + 3*x^3*y^2*z^2 + 2*x^2*y^3*z^2 - x^4*z^3 + 2*x^3*y*z^3 + 2*x^2*y^2*z^3 + y^4*z^3 + 3*x^3*z^4 - 2*x^2*y*z^4 - x*y^2*z^4 - y^3*z^4 + 3*x^2*z^5 - 2*x*y*z^5 - 3*y^2*z^5 - 3*y*z^6 + 3*z^7 - 18047/148176*x^6 + x^4*y^2 - x^3*y^3 + 3*y^6 - x^5*z + 2*x^4*y*z - 2*x^3*y^2*z + 2*x^2*y^3*z + 3*x*y^4*z + 3*y^5*z + 3*x^4*z^2 - x^3*y*z^2 - 2*x^2*y^2*z^2 + x*y^3*z^2 - 3*y^4*z^2 + 2*x^3*z^3 + x^2*y*z^3 + 2*x*y^2*z^3 + y^3*z^3 + 2*x^2*z^4 + x*y*z^4 + 3*y^2*z^4 + x*z^5 + 3*y*z^5 + 2*z^6 + 23/882*x^5 - 3*x^4*y - 3*x^3*y^2 - x^2*y^3 + 2*x*y^4 - 2*y^5 + 17/12*x^4*z + x^3*y*z - 2*x^2*y^2*z + 3*x*y^3*z - 2*y^4*z - x^3*z^2 - 3*x^2*y*z^2 + x*y^2*z^2 + y^3*z^2 - 2*x^2*z^3 + x*y*z^3 + 2*y^2*z^3 + 3*x*z^4 - 2*z^5 + x^3*y + x*y^3 + 3*y^4 - 1/6*x^3*z + 2*x^2*y*z - x*y^2*z - 2*y^3*z + 2*x^2*z^2 + 3*y^2*z^2 + 3*x*z^3 + 2*z^4 + x*y^2 + y^3 - 2*x*y*z + 2*y^2*z - 3*y*z^2']
  ok   R4 N7, component W4 (P_402 = P_203 = 0, 5P_501^2 = 21P_302): a point of the component (residual equations vanish)
      timing: member+triples 0.5 s, stabilization 0.3 s
  ok   R4 N7, component W4 (P_402 = P_203 = 0, 5P_501^2 = 21P_302) anchor (A): fixed {'P_402': 0, 'P_203': 0}, solved ['P_302']; member closed with H and the prescribed graded data: True; a triple of I of local length 37 (ds), stabilization (37, 12)
      anchor generators: {'P': 'X^7 + 2*X^5*Z + 20/21*X^3*Z^2 + 2*X*Y^2 - 2*Y^3 + 2*X*Y*Z - Y^2*Z - Y*Z^2 - Z^3', 'G': 'X^4*Z - X^2*Y^2 - 2*X*Y^2 + Y^3 + X*Y*Z + 2*Z^3', 'H': 'X^3*Y - 3/2*X^2*Z^2 - 2*X*Y^2 - Y^3 - Y^2*Z - Z^3'}
      anchor triple: ['-3*x^8 - x^7*y + 3*x^6*y^2 - 2*x^5*y^3 - 3*x^4*y^4 + 2*x^2*y^6 - 3*x*y^7 + 3*x^7*z + 3*x^6*y*z - x^4*y^3*z - 3*x^3*y^4*z + 3*x^2*y^5*z - 3*x*y^6*z - y^7*z + 3*x^6*z^2 - 2*x^4*y^2*z^2 + x^3*y^3*z^2 - y^6*z^2 - x^5*z^3 + x^4*y*z^3 + x^3*y^2*z^3 + 3*x*y^4*z^3 - y^5*z^3 + x^4*z^4 + x^3*y*z^4 - 3*x^2*y^2*z^4 - 3*x*y^3*z^4 + y^4*z^4 + 2*x^3*z^5 - 2*x^2*y*z^5 - 3*x*y^2*z^5 + 2*y^3*z^5 + 3*x^2*z^6 - x*y*z^6 - 3*y^2*z^6 - 2*x*z^7 + 3*y*z^7 - 3*z^8 + 62/441*x^7 + 2*x^5*y^2 - 3*x^3*y^4 - x*y^6 - y^7 + 3*x^6*z - 2*x^5*y*z - 2*x^2*y^4*z - 3*x*y^5*z + 3*y^6*z + 2*x^5*z^2 + 2*x^4*y*z^2 + 3*x^3*y^2*z^2 - 3*x^2*y^3*z^2 - 3*x*y^4*z^2 - 2*y^5*z^2 - 2*x^4*z^3 - 2*x^3*y*z^3 + x^2*y^2*z^3 - 3*y^4*z^3 + 3*x^3*z^4 + 2*x^2*y*z^4 - 2*x*y^2*z^4 - x^2*z^5 - x*y*z^5 + 2*y^2*z^5 - x*z^6 + 3*y*z^6 + 2*z^7 - 37/3528*x^6 - 2*x^5*y - 3*x^4*y^2 + 2*x^3*y^3 + x^2*y^4 + 2*x*y^5 - 3*x^5*z + 2*x^4*y*z + 3*x^3*y^2*z + x^2*y^3*z + 3*x*y^4*z + 3*y^5*z - x^4*z^2 - 2*x^3*y*z^2 - x*y^3*z^2 + 3*y^4*z^2 - 2*x^3*z^3 - 3*x^2*y*z^3 + 2*x*y^2*z^3 - y^3*z^3 - 2*x^2*z^4 + 3*y^2*z^4 + 3*x*z^5 + z^6 - 1/63*x^5 - 2*x^4*y - x^3*y^2 - 3*x^2*y^3 - 3*y^5 + 1/8*x^4*z - x*y^3*z + y^4*z + x^3*z^2 + x^2*y*z^2 + x*y^2*z^2 + 2*y^3*z^2 + 2*x^2*z^3 - 2*x*y*z^3 - 3*x*z^4 + y*z^4 - 3*z^5 + 7/3*x^3*y - 3*x^2*y^2 - 2*y^4 + 1/3*x^3*z - 3*x^2*y*z + 3*x*y^2*z + y^3*z + 2*x^2*z^2 - x*y*z^2 - x*z^3 - y*z^3 - 3*z^4 + 2*x*y^2 + y^3 - x*y*z - 3*y^2*z + 3*y*z^2 - z^3', 'x^8 - 3*x^7*y - x^5*y^3 - 3*x^4*y^4 - 2*x^3*y^5 - 3*x^2*y^6 + 3*x*y^7 - 2*y^8 + 2*x^7*z - 3*x^6*y*z + 3*x^5*y^2*z - 2*x^4*y^3*z + x^3*y^4*z + 2*x^2*y^5*z - 3*x*y^6*z - 3*y^7*z + 3*x^6*z^2 + x^5*y*z^2 + 2*x^4*y^2*z^2 - x^3*y^3*z^2 + 3*x^2*y^4*z^2 + 3*x*y^5*z^2 - y^6*z^2 + 2*x^5*z^3 + 3*x^4*y*z^3 - 3*x^2*y^3*z^3 - x*y^4*z^3 + 2*y^5*z^3 - 2*x^3*y*z^4 - 3*x^2*y^2*z^4 - 3*x*y^3*z^4 + 2*y^4*z^4 - x^3*z^5 - 3*x^2*y*z^5 + x*y^2*z^5 - y^3*z^5 + 2*x^2*z^6 + 3*x*y*z^6 + y^2*z^6 - 2*x*z^7 - y*z^7 - 3*z^8 + 2419/17640*x^7 + 2*x^6*y + x^5*y^2 + 2*x^4*y^3 + 3*x^3*y^4 - 2*x^2*y^5 + 2*x*y^6 + 3*y^7 - x^6*z - x^5*y*z - 2*x^4*y^2*z - x^3*y^3*z - 2*x*y^5*z + x^5*z^2 - x^4*y*z^2 - 2*x^2*y^3*z^2 + 3*x*y^4*z^2 + x^4*z^3 + x^3*y*z^3 + 3*x*y^3*z^3 - 2*y^4*z^3 - 2*x^3*z^4 + 3*x^2*y*z^4 - 2*x*y^2*z^4 - 2*y^3*z^4 - 2*x^2*z^5 + 3*x*y*z^5 + 3*y^2*z^5 - x*z^6 + 3*y*z^6 + 3*z^7 + 185/3528*x^6 - 2*x^4*y^2 + 3*x^3*y^3 - 3*x^2*y^4 - x*y^5 - 3*x^5*z - 2*x^4*y*z + x^2*y^3*z - x*y^4*z - x^4*z^2 + 3*x^3*y*z^2 - x^2*y^2*z^2 + y^4*z^2 + 2*x^2*y*z^3 - 2*x*y^2*z^3 + y^3*z^3 - x^2*z^4 + x*y*z^4 - y^2*z^4 + 2*y*z^5 + 2*z^6 + 1/63*x^5 - x^4*y - 3*x^3*y^2 + x*y^4 - 23/24*x^4*z - 3*x^3*y*z - 3*x^2*y^2*z + 3*x*y^3*z - 2*y^4*z + 2*x^3*z^2 + x^2*y*z^2 - 3*x*y^2*z^2 - 2*y^3*z^2 + 3*x*y*z^3 - 3*y^2*z^3 - 2*x*z^4 + 2*y*z^4 - z^5 - 14/3*x^3*y - 3*x^2*y^2 - x*y^3 - 1/3*x^3*z - 3*x^2*y*z - 3*x*y^2*z + 2*y^3*z - 3*x^2*z^2 + 3*y^2*z^2 - 3*x*z^3 + 3*y*z^3 - 3*z^4 - 2*x*y^2 - 2*y^3 + 3*x*y*z + 2*y^2*z + 3*y*z^2 + z^3', '3*x^8 + 3*x^7*y - x^6*y^2 - 2*x^5*y^3 - 3*x^4*y^4 + 3*x^3*y^5 + 2*x^2*y^6 + x*y^7 - 2*y^8 - 3*x^6*y*z - 2*x^5*y^2*z - 2*x^4*y^3*z - x^2*y^5*z + 2*x*y^6*z + 2*y^7*z + 2*x^6*z^2 + x^4*y^2*z^2 - 2*x^3*y^3*z^2 - 2*x*y^5*z^2 - 2*y^6*z^2 + x^5*z^3 + 3*x^4*y*z^3 - 2*x^3*y^2*z^3 + 2*x^2*y^3*z^3 + x*y^4*z^3 - 3*y^5*z^3 + 3*x^4*z^4 - 3*x^3*y*z^4 + 2*x^2*y^2*z^4 - 2*x*y^3*z^4 - 2*y^4*z^4 - 2*x^3*z^5 + 3*x*y^2*z^5 - 2*y^3*z^5 + 2*x^2*z^6 - 2*x*y*z^6 + y^2*z^6 + 3*x*z^7 + 3*y*z^7 - z^8 + 157/1960*x^7 - x^6*y - 3*x^5*y^2 - 2*x^4*y^3 + 2*x^2*y^5 + x*y^6 + 3*y^7 - x^6*z + 3*x^5*y*z - x^4*y^2*z + x^3*y^3*z + 2*x*y^5*z - y^6*z + 3*x^5*z^2 - x^4*y*z^2 - x^3*y^2*z^2 + 2*x^2*y^3*z^2 + x*y^4*z^2 - 2*x^4*z^3 + 2*x^3*y*z^3 + x*y^3*z^3 - y^4*z^3 - 3*x^3*z^4 + x^2*y*z^4 - 2*x*y^2*z^4 + y^3*z^4 - x^2*z^5 + x*y*z^5 - y^2*z^5 + 2*x*z^6 + y*z^6 - 2*z^7 - 1/294*x^6 - x^5*y + 2*x^4*y^2 - x^3*y^3 + 3*x^2*y^4 + 2*x*y^5 + y^6 - 2*x^5*z - 2*x^4*y*z + 2*x^3*y^2*z + 3*x^2*y^3*z + 3*x*y^4*z + 3*y^5*z - 2*x^3*y*z^2 + 3*x^2*y^2*z^2 - 3*x*y^3*z^2 - 3*y^4*z^2 + 3*x^3*z^3 + 2*x*y^2*z^3 + x^2*z^4 + 2*x*y*z^4 + 2*y^2*z^4 - 2*x*z^5 + 2*y*z^5 - 2*z^6 - 1/126*x^5 + 2*x^4*y - x^3*y^2 - 2*x^2*y^3 - 2*x*y^4 + 2*y^5 + 1/6*x^4*z - 2*x^3*y*z + 3*x^2*y^2*z + 2*x*y^3*z - y^4*z + 3*x^3*z^2 + x^2*y*z^2 - 3*x*y^2*z^2 + 2*y^3*z^2 - 2*x^2*z^3 - 3*x*y*z^3 - 3*y^2*z^3 - x*z^4 + 2*y*z^4 - 13/3*x^3*y - x^2*y^2 - 2*x*y^3 + 2*y^4 + 1/6*x^3*z + x^2*y*z - 3*x*y^2*z + 3*y^3*z - 2*x^2*z^2 + 2*x*y*z^2 + y^2*z^2 + x*z^3 + 3*y*z^3 - 2*z^4 + x*y^2 - 3*y^3 + 2*x*y*z - 3*y^2*z + y*z^2 + z^3']
--- R4 N1
  ok   R4 N1: the graded normal form is closed with H = [1, 3, 6, 4, 3, 2, 1, 1]
  ok   R4 N1: one lift per pivot of D_4..D_7
    134 closedness conditions in 88 tail coefficients; 56 eliminated with constant pivots; residual equations:
       (-15/4) * G_130 * (2*P_501 - 7*G_211)
       (315/2) * G_103 * G_130
       (-1/588) * (-20*P_510^2*P_501 + 70*P_510^2*G_211 - 2205*P_510*G_301*G_130 + 15435*G_301*G_220*G_130 + 84*P_501*P_320 - 294*P_320*G_211 + 1764*P_401*G_130 - 15435*G_130*G_111 + 1764*P_203*H_120)
       (-1/168) * (20*P_510*P_501*G_130 + 140*P_501*G_220*G_130 - 70*P_510*G_211*G_130 + 2205*G_301*G_130^2 - 2940*G_130*G_021 + 504*P_203*H_030)
       (-1/4) * (5*P_501*G_211*G_130 - 210*G_130*G_012 + 12*P_203*H_021)
       (-1/14) * (5*P_510^2*G_103 - 2*P_510*P_203 - 21*P_320*G_103 - 2205*G_130*G_003 + 42*P_203*H_012)
       (-15/14) * G_130 * (2*P_501 - 7*G_211)
       (45/2) * G_103 * G_130
  ok   R4 N1: on {G_130 != 0} the residual equations are triangular in ['G_211', 'G_103', 'G_111', 'G_021', 'G_012', 'G_003'] with diagonal (const)*G_130: a graph (irreducible)
  ok   R4 N1: on {G_130 = 0}: on {P_203 != 0} triangular in ['H_120', 'H_030', 'H_021', 'H_012'] with diagonal (const)*P_203; on {P_203 = 0}: c*(linear forms), c = -5*P_510^2 + 21*P_320: ['-1/147*P_501 + 1/42*G_211', '1/14*G_103']
  ok   R4 N1, component W1 (G_130 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N1, component W1 (G_130 != 0) anchor (A): fixed {'G_130': 1}, solved ['G_211', 'G_103', 'G_111', 'G_021', 'G_012', 'G_003']; member closed with H and the prescribed graded data: True; a triple of I of local length 36 (ds), stabilization (36, 11)
      anchor generators: {'P': 'X^7 + 105/4*X^4*Y^2 + 105/8*X*Y^4 + 35/2*X*Y^3 - 105/4*Y^4 - 2*X*Y^2 - 2*X*Y*Z + 2*Y^2*Z - Z^3', 'G': 'X^4*Y + X*Y^3 + X*Y^2 - 2*Y^3', 'H': 'X^2*Z^2 + 2*X*Y^2 - 2*X*Y*Z - 2*Y^2*Z - 2*Y*Z^2'}
      anchor triple: ['x^6*y^2 - 2*x^5*y^3 + 2*x^4*y^4 + x^3*y^5 - 3*y^8 - x^7*z - 2*x^6*y*z - 3*x^5*y^2*z + 2*x^3*y^4*z + 3*x^2*y^5*z + 3*x*y^6*z + y^7*z - 3*x^6*z^2 - 2*x^5*y*z^2 - 2*x^4*y^2*z^2 + x^3*y^3*z^2 + 3*x^2*y^4*z^2 - 3*x*y^5*z^2 + x^5*z^3 - x^4*y*z^3 + x^3*y^2*z^3 + 3*x^2*y^3*z^3 + 3*x*y^4*z^3 - 2*y^5*z^3 + x^4*z^4 - x^3*y*z^4 - x^2*y^2*z^4 - 2*x*y^3*z^4 - y^4*z^4 - 2*x^3*z^5 - 2*x*y^2*z^5 - 2*x^2*z^6 - 3*x*y*z^6 - 2*y^2*z^6 - 2*x*z^7 - z^8 - 893/1260*x^7 - x^6*y + 3*x^5*y^2 + 2*x^4*y^3 - 2*x^3*y^4 - 3*x^2*y^5 - x*y^6 + 2*y^7 + 3*x^5*y*z - 2*x^4*y^2*z - 3*x^3*y^3*z + 3*x^2*y^4*z + 2*x*y^5*z + 3*y^6*z - 3*x^4*y*z^2 - 3*x^3*y^2*z^2 - 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 - 3*x^4*z^3 - 2*x^3*y*z^3 - 3*x^2*y^2*z^3 + 3*y^4*z^3 - x^3*z^4 + 3*x^2*y*z^4 + 3*x*y^2*z^4 - y^3*z^4 + 3*x^2*z^5 - x*y*z^5 + x*z^6 - 9/16*x^6 - x^5*y + 2*x^4*y^2 - 3*x*y^5 - 2*y^6 - 2*x^5*z - x^4*y*z + 2*x^3*y^2*z + 2*x*y^4*z + 3*x^4*z^2 + 3*x^3*y*z^2 + 3*x^2*y^2*z^2 + 3*x*y^3*z^2 + y^4*z^2 - 3*x^3*z^3 + x^2*y*z^3 + x*y^2*z^3 + 2*y^3*z^3 + 2*x^2*z^4 + 2*x*y*z^4 + 2*y^2*z^4 + x*z^5 + 2*y*z^5 - 2*z^6 + 1/4*x^5 + 1/2*x^4*y + 3*x^3*y^2 + x^2*y^3 - 2*x*y^4 - 3*y^5 - 2*x^4*z - x^3*y*z - 2*x^2*y^2*z + 2*x*y^3*z - 2*y^4*z - x^3*z^2 + 2*x^2*y*z^2 + 2*x*y^2*z^2 - 2*x^2*z^3 + 3*x*y*z^3 - y^2*z^3 - 3*x*z^4 - y*z^4 + z^5 - x^2*y^2 - 2*x*y^3 - 3*y^4 - 3*x^3*z + 3*x^2*y*z - x*y^2*z - y^3*z - 3*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + 2*x*z^3 + 3*y*z^3 + z^4 - y^2*z - 2*y*z^2 - z^3', 'x^8 + x^6*y^2 + 2*x^5*y^3 + 2*x^3*y^5 + 3*x^2*y^6 - 3*x*y^7 + 3*x^7*z - x^6*y*z + 3*x^5*y^2*z - x^4*y^3*z - 3*x^3*y^4*z - 3*x^2*y^5*z + 3*x*y^6*z + x^6*z^2 - 2*x^4*y^2*z^2 + 3*x^3*y^3*z^2 + 2*x^2*y^4*z^2 + 3*x*y^5*z^2 - 2*y^6*z^2 + 3*x^5*z^3 + 2*x^4*y*z^3 - x^3*y^2*z^3 - 2*x^2*y^3*z^3 - x*y^4*z^3 + 2*y^5*z^3 + 3*x^4*z^4 + 3*x^3*y*z^4 + x^2*y^2*z^4 - 3*x*y^3*z^4 - 3*y^4*z^4 + x^3*z^5 - 2*x^2*y*z^5 - 3*x*y^2*z^5 + 2*y^3*z^5 + 3*x^2*z^6 + x*y*z^6 + 2*y^2*z^6 - x*z^7 - 2*z^8 - 1789/5040*x^7 + 2*x^6*y - 3*x^5*y^2 - 3*x^4*y^3 + 3*x^3*y^4 - x^2*y^5 - 3*x*y^6 - 3*y^7 + 2*x^6*z - x^5*y*z - 3*x^4*y^2*z - 3*x^3*y^3*z - x^2*y^4*z - x*y^5*z - 3*y^6*z + x^5*z^2 - x^4*y*z^2 + x^2*y^3*z^2 - 3*x*y^4*z^2 + 3*y^5*z^2 + 3*x^4*z^3 + x^3*y*z^3 - x^2*y^2*z^3 - y^4*z^3 - 3*x^2*y*z^4 + x*y^2*z^4 + 3*y^3*z^4 - x^2*z^5 - 3*x*y*z^5 + 2*x*z^6 - 3*y*z^6 - 3*z^7 + 1/2*x^6 + 3*x^5*y + x^4*y^2 - 3*x^3*y^3 - 3*x^2*y^4 - 3*x*y^5 + x^5*z + x^4*y*z - 2*x^2*y^3*z + 3*y^5*z - 3*x^4*z^2 - x^3*y*z^2 - 2*x^2*y^2*z^2 - 3*x*y^3*z^2 + 3*y^4*z^2 + x^3*z^3 - 2*x^2*y*z^3 - 2*x*y^2*z^3 + 3*y^3*z^3 - 3*x^2*z^4 + 2*x*y*z^4 - 2*y^2*z^4 - 2*x*z^5 + 3*y*z^5 - 3*z^6 - 3/4*x^5 + 11/6*x^4*y - 2*x^3*y^2 + 2*x^2*y^3 - y^5 + 2*x^4*z + x^2*y^2*z + 2*x*y^3*z + y^4*z - 2*x^3*z^2 - 2*x^2*y*z^2 - x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 - 3*x*y*z^3 - 2*y^2*z^3 + 2*x*z^4 + 2*y*z^4 + 3*z^5 + 1/4*x^4 - 3/4*x^3*y + 3*x^2*y^2 - x*y^3 - y^4 + 3*x^3*z - 3*x^2*y*z + x*y^2*z + 3*y^3*z + 9/2*x^2*z^2 - 3*x*y*z^2 + 3*y^2*z^2 - 3*x*z^3 + 3*y*z^3 - x*y^2 + 3*y^3 + x*y*z + 2*y^2*z + y*z^2 + z^3', 'x^8 + x^7*y - 2*x^6*y^2 - x^5*y^3 + 2*x^4*y^4 - 3*x^3*y^5 + 2*x^2*y^6 - 2*x*y^7 + 3*y^8 + 3*x^7*z - 2*x^5*y^2*z - 3*x^4*y^3*z - x^3*y^4*z - 3*x^2*y^5*z + 3*x*y^6*z - y^7*z - 2*x^6*z^2 + 3*x^5*y*z^2 - x^3*y^3*z^2 + 3*x^2*y^4*z^2 - x*y^5*z^2 - y^6*z^2 - 3*x^5*z^3 + 3*x^4*y*z^3 + 3*x^3*y^2*z^3 - x^2*y^3*z^3 - 3*x*y^4*z^3 + y^5*z^3 + 3*x^4*z^4 + x^3*y*z^4 + x*y^3*z^4 - 2*y^4*z^4 - 3*x^3*z^5 + x^2*y*z^5 + 2*x*y^2*z^5 + 3*x^2*z^6 + 3*x*y*z^6 + 2*y^2*z^6 - 3*x*z^7 - y*z^7 + z^8 - 1357/5040*x^7 + 3*x^6*y + 3*x^5*y^2 - 3*x^4*y^3 + 2*x^3*y^4 - x^2*y^5 + x*y^6 - 2*y^7 - 3*x^6*z + 3*x^5*y*z - x^4*y^2*z + 3*x^2*y^4*z - 2*x*y^5*z + 2*x^5*z^2 - 2*x^3*y^2*z^2 + 3*x^2*y^3*z^2 - 3*x*y^4*z^2 - 2*y^5*z^2 + 2*x^4*z^3 + 3*x^3*y*z^3 - 3*x^2*y^2*z^3 - x*y^3*z^3 + 3*y^4*z^3 - 3*x^3*z^4 + 3*x^2*y*z^4 - 2*x*y^2*z^4 - 2*y^3*z^4 - 2*x*y*z^5 - y^2*z^5 - 2*x*z^6 + 3*z^7 + 9/16*x^6 - x^5*y + 2*x^4*y^2 + 3*x^3*y^3 - 2*x^2*y^4 + 3*x*y^5 + 3*y^6 + x^5*z + 2*x^4*y*z - 2*x^3*y^2*z - 3*x^2*y^3*z - 3*x^3*y*z^2 + 2*x^2*y^2*z^2 + 2*y^4*z^2 + 2*x^3*z^3 - x^2*y*z^3 + 2*x*y^2*z^3 + 3*y^3*z^3 + x^2*z^4 + 3*x*y*z^4 - 2*y^2*z^4 + z^6 - 3/4*x^5 + 2*x^4*y - 3*x^3*y^2 + 3*x^2*y^3 + x*y^4 - x^2*y^2*z + 3*x*y^3*z + y^4*z - 3*x^3*z^2 + x^2*y*z^2 - x*y^2*z^2 + 3*y^3*z^2 - 2*x*y*z^3 - 2*y^2*z^3 + 3*y*z^4 + 2*z^5 - 3/4*x^3*y + 3*x^2*y^2 - 2*x*y^3 + 2*y^4 + x^3*z - 3*x^2*y*z - x*y^2*z - 2*y^3*z + 5/2*x^2*z^2 - 3*y^2*z^2 + 3*x*z^3 + y*z^3 + 2*z^4 + 3*y^3 + 3*x*y*z - 2*y^2*z + 3*y*z^2 - z^3']
  ok   R4 N1, component W2 (G_130 = 0, P_203 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N1, component W2 (G_130 = 0, P_203 != 0) anchor (A): fixed {'G_130': 0, 'P_203': 1}, solved ['H_120', 'H_030', 'H_021', 'H_012']; member closed with H and the prescribed graded data: True; a triple of I of local length 36 (ds), stabilization (36, 11)
      anchor generators: {'P': 'X^7 + X^2*Z^3 - 3*X*Y*Z^2 - 3/4*Z^4 - X*Y^2 - Y^3 + X*Y*Z + Y^2*Z - 2*Y*Z^2 + 2*Z^3', 'G': 'X^4*Y + X*Y^2 + 2*Y^3 + 2*X*Y*Z + 2*Y^2*Z - Z^3', 'H': 'X^2*Z^2 - 2*X*Y*Z - Z^3'}
      anchor triple: ['-2*x^8 + 2*x^7*y + 3*x^6*y^2 + 3*x^5*y^3 + 2*x^4*y^4 + 2*x^2*y^6 + 3*x*y^7 - 3*y^8 + 3*x^7*z + 3*x^5*y^2*z - 2*x^4*y^3*z + 2*x^3*y^4*z + 3*x^2*y^5*z - 3*x*y^6*z + 3*y^7*z - 3*x^6*z^2 + 2*x^4*y^2*z^2 - 2*x^3*y^3*z^2 - 3*x^2*y^4*z^2 - 3*x*y^5*z^2 + 3*y^6*z^2 + x^5*z^3 - 2*x^3*y^2*z^3 + x^2*y^3*z^3 + 3*x*y^4*z^3 - y^5*z^3 + 3*x^4*z^4 + x^3*y*z^4 + x^2*y^2*z^4 + x*y^3*z^4 + y^4*z^4 + x^3*z^5 + 2*x^2*y*z^5 - 2*x*y^2*z^5 + 3*y^3*z^5 - x^2*z^6 + y^2*z^6 - x*z^7 + 3*y*z^7 - z^8 + 1/144*x^7 + 3*x^6*y - 3*x^5*y^2 + x^4*y^3 + 2*x^3*y^4 - 3*x^2*y^5 + x*y^6 - y^7 + x^6*z + 3*x^5*y*z + 3*x^4*y^2*z + 3*x^3*y^3*z - x^2*y^4*z - x*y^5*z + 3*y^6*z + 2*x^4*y*z^2 - 2*x^3*y^2*z^2 - 2*x^2*y^3*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 + 2*x^4*z^3 - x^3*y*z^3 + 2*x^2*y^2*z^3 - 3*x*y^3*z^3 + 3*y^4*z^3 - 2*x^3*z^4 - x^2*y*z^4 + 3*x*y^2*z^4 - y^3*z^4 + 2*x^2*z^5 + 3*x*y*z^5 - 2*y^2*z^5 + x*z^6 - y*z^6 - 3*z^7 + 1/840*x^6 - 3*x^5*y + x^4*y^2 + x^3*y^3 - 3*x^2*y^4 + x*y^5 + 3*y^6 + x^5*z - 3*x^4*y*z - x^3*y^2*z + x^2*y^3*z - x*y^4*z + 3*y^5*z + x^4*z^2 - x^3*y*z^2 - x^2*y^2*z^2 + x*y^3*z^2 - 3*y^4*z^2 + 3*x^3*z^3 - 3*y^3*z^3 - 2*x^2*z^4 - 3*y^2*z^4 - x*z^5 - y*z^5 + z^6 + 1/140*x^5 - 1/12*x^4*y + 3*x^3*y^2 + 3*x^2*y^3 - x*y^4 + 3*y^5 + 3*x^4*z + 2*x^3*y*z + 3*x^2*y^2*z + 2*x*y^3*z + 3*y^4*z + 3*x^3*z^2 - 3*x^2*y*z^2 - x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 + 3*x*y*z^3 - 2*y^2*z^3 - 3*x*z^4 - y*z^4 - 2*z^5 - x^2*y^2 + 3*x*y^3 - 2*y^4 + 2*x^3*z - x^2*y*z - x*y^2*z + y^3*z - 5*x^2*z^2 - y^2*z^2 - 2*x*z^3 + 3*y*z^3 + 2*z^4 - 3*x*y^2 - y^3 - x*y*z + y^2*z - 3*y*z^2 - 3*z^3', '-2*x^8 + 2*x^7*y + x^6*y^2 - 3*x^4*y^4 - 2*x^3*y^5 - 2*x^2*y^6 + 2*x*y^7 - y^8 - x^7*z - 3*x^6*y*z + 3*x^4*y^3*z + 2*x^3*y^4*z + x^2*y^5*z + 2*x^5*y*z^2 - x^4*y^2*z^2 + 3*x^3*y^3*z^2 - 2*x^2*y^4*z^2 + 2*x*y^5*z^2 + 3*y^6*z^2 - x^5*z^3 - 3*x^4*y*z^3 - 3*x^3*y^2*z^3 + x^2*y^3*z^3 + 2*x*y^4*z^3 - y^5*z^3 - 3*x^4*z^4 + 3*x^2*y^2*z^4 - x*y^3*z^4 - y^4*z^4 - 3*x^3*z^5 - 3*x^2*y*z^5 + 2*x*y^2*z^5 + y^3*z^5 + x^2*z^6 + x*y*z^6 + 2*x*z^7 + 3*y*z^7 + 3*z^8 - 1/48*x^7 - x^6*y + 2*x^5*y^2 - 3*x^4*y^3 + 2*x^3*y^4 - 3*x^2*y^5 + 3*y^7 + x^6*z - 2*x^5*y*z + 3*x^4*y^2*z - 3*x^3*y^3*z + 2*x^2*y^4*z + 2*x*y^5*z + 2*y^6*z + 2*x^5*z^2 + x^4*y*z^2 - 3*x^3*y^2*z^2 + x^2*y^3*z^2 + 3*x*y^4*z^2 - x^3*y*z^3 - x^2*y^2*z^3 + 2*x*y^3*z^3 - 3*y^4*z^3 - x^3*z^4 - y^3*z^4 - 2*x^2*z^5 - 3*x*y*z^5 + 2*y^2*z^5 - 2*x*z^6 - 2*y*z^6 - z^7 + 1/840*x^6 + 3*x^5*y - 3*x^4*y^2 + x^3*y^3 - y^6 + 3*x^5*z + 2*x^3*y^2*z - 2*x^2*y^3*z - 3*x*y^4*z + y^5*z - 2*x^4*z^2 + 2*x^3*y*z^2 - 3*x^2*y^2*z^2 - x*y^3*z^2 - 2*y^4*z^2 - 2*x^3*z^3 + 3*x^2*y*z^3 - x*y^2*z^3 - x^2*z^4 - 2*x*y*z^4 - 2*y^2*z^4 - 3*x*z^5 - 3*y*z^5 + z^6 - 1/140*x^5 + 13/12*x^4*y + 2*x^3*y^2 - x*y^4 - y^5 - x^4*z + x^3*y*z - 3*x^2*y^2*z - 3*y^4*z - 3*x^3*z^2 - x*y^2*z^2 - 2*y^3*z^2 + 3*x^2*z^3 - x*y*z^3 + 2*y^2*z^3 - 3*x*z^4 + y*z^4 + 3*z^5 - 3*x^2*y^2 + 3*x*y^3 + 2*y^4 - x^3*z + x^2*y*z - 2*x*y^2*z + 3*y^3*z + 3*x^2*z^2 - 3*y^2*z^2 - x*z^3 - 3*y*z^3 - 2*z^4 - 3*x*y^2 + y^3 - 3*x*y*z - 2*y^2*z - y*z^2 + 3*z^3', '-x^8 + 3*x^6*y^2 - 2*x^5*y^3 - 3*x^4*y^4 + 2*x^3*y^5 - 3*x*y^7 - 3*x^7*z - x^5*y^2*z + x^4*y^3*z + x^3*y^4*z + 3*x^2*y^5*z - x*y^6*z - 3*y^7*z + 3*x^6*z^2 - x^5*y*z^2 - 2*x^3*y^3*z^2 - 2*x^2*y^4*z^2 + x*y^5*z^2 + y^6*z^2 - 3*x^5*z^3 - x^3*y^2*z^3 - x^2*y^3*z^3 - 2*x*y^4*z^3 + 2*y^5*z^3 + 2*x^4*z^4 + 2*x^3*y*z^4 + 3*x^2*y^2*z^4 - x*y^3*z^4 - 3*y^4*z^4 - x^3*z^5 - 2*x^2*y*z^5 + x*y^2*z^5 - x^2*z^6 + 3*x*y*z^6 - 2*y^2*z^6 + 2*x*z^7 - 3*y*z^7 + 1/1680*x^7 - x^6*y - 2*x^5*y^2 - 2*x^4*y^3 + 3*x^3*y^4 - 3*x^2*y^5 - 2*x*y^6 - 2*y^7 - 2*x^6*z + 3*x^5*y*z + 3*x^4*y^2*z + x^3*y^3*z + 2*x^2*y^4*z + 2*x*y^5*z + 2*y^6*z - x^5*z^2 + 2*x^4*y*z^2 - 2*x^2*y^3*z^2 - 3*x*y^4*z^2 - 2*y^5*z^2 + 3*x^3*y*z^3 + 3*x^2*y^2*z^3 + x*y^3*z^3 + 2*y^4*z^3 + 2*x^3*z^4 + 2*x^2*y*z^4 + 3*x*y^2*z^4 + y^3*z^4 - 3*x^2*z^5 - x*y*z^5 + 2*y^2*z^5 - 2*x*z^6 - 3*z^7 + 1/168*x^6 - x^5*y + 3*x^4*y^2 - 2*x^3*y^3 + x^2*y^4 - x*y^5 - 2*y^6 + 3*x^4*y*z + 2*x^3*y^2*z + x^2*y^3*z + x*y^4*z - y^5*z - x^4*z^2 + 3*x^3*y*z^2 - x^2*y^2*z^2 + 3*y^4*z^2 - x^2*y*z^3 + 3*x*y^2*z^3 + y^3*z^3 - 3*x^2*z^4 + 2*x*y*z^4 + 2*y^2*z^4 + 3*y*z^5 - 1/420*x^5 - 4/3*x^4*y + 2*x^3*y^2 + x^2*y^3 + x*y^4 - 3*y^5 + 3*x^4*z + 2*x^3*y*z - 2*x^2*y^2*z - 2*y^4*z - 2*x^3*z^2 - 3*x^2*y*z^2 - 2*x*y^2*z^2 + 3*y^3*z^2 - 2*x^2*z^3 - x*y*z^3 + 3*y^2*z^3 + 2*x*z^4 + 3*z^5 - x^2*y^2 - 3*x*y^3 - 3*y^4 - 2*x^3*z + x^2*y*z - 3*x*y^2*z + y^3*z + 3*x^2*z^2 - x*y*z^2 + 3*y^2*z^2 - 2*x*z^3 - 3*y*z^3 - z^4 - 2*x*y^2 + 3*y^3 + 3*x*y*z + y*z^2 + z^3']
  ok   R4 N1, component W3 (G_130 = P_203 = 0, c = 0): a point of the component (residual equations vanish)
      timing: member+triples 0.6 s, stabilization 0.3 s
  ok   R4 N1, component W3 (G_130 = P_203 = 0, c = 0) anchor (A): fixed {'G_130': 0, 'P_203': 0}, solved ['P_320']; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 13)
      anchor generators: {'P': 'X^7 - X*Y^2 - 2*Y^3 - X*Y*Z - 2*Y*Z^2 - 2*Z^3', 'G': 'X^4*Y - 2*X^2*Y*Z - X*Z^3 + X*Y*Z + Z^3', 'H': 'X^2*Z^2 + X*Y^2 - Y^3 - 2*X*Y*Z - 2*Y^2*Z'}
      anchor triple: ['x^8 + 2*x^7*y - x^6*y^2 + x^5*y^3 + 2*x^4*y^4 - x^3*y^5 - x^2*y^6 - x*y^7 - 3*y^8 + 2*x^7*z - x^6*y*z + x^5*y^2*z + 2*x^4*y^3*z - x^3*y^4*z + 3*x^2*y^5*z + x*y^6*z - 2*y^7*z + x^6*z^2 + x^4*y^2*z^2 + 3*x^3*y^3*z^2 - x^2*y^4*z^2 - 3*x*y^5*z^2 + y^6*z^2 - x^5*z^3 - x^4*y*z^3 + x^3*y^2*z^3 - 2*x^2*y^3*z^3 + x*y^4*z^3 - 2*x^2*y^2*z^4 + x*y^3*z^4 + 2*y^4*z^4 + 2*x^3*z^5 + 3*x^2*y*z^5 + 2*y^3*z^5 - 2*x*y*z^6 - 3*y^2*z^6 + x*z^7 - 2*y*z^7 - 2*z^8 - 1/840*x^7 + x^5*y^2 + x^4*y^3 + x^3*y^4 - x^2*y^5 - 2*x*y^6 + 2*y^7 - 3*x^6*z - x^5*y*z + x^2*y^4*z - 2*x*y^5*z + 3*y^6*z + x^5*z^2 + x^4*y*z^2 + 2*x^3*y^2*z^2 - 3*x^2*y^3*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 + 3*x^4*z^3 - x^2*y^2*z^3 - 2*x*y^3*z^3 + 3*x^3*z^4 - 2*x^2*y*z^4 - x*y^2*z^4 + 3*y^3*z^4 + x*y*z^5 + 2*y^2*z^5 - x*z^6 + 2*y*z^6 - z^7 - 3*x^5*y + 3*x^4*y^2 + 3*x^3*y^3 + 3*x^2*y^4 + x*y^5 - 3*y^6 + 3*x^5*z + 3*x^4*y*z - x^2*y^3*z - 3*x*y^4*z - 2*y^5*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 + 2*y^4*z^2 + x^3*z^3 - 3*y^3*z^3 - x*y*z^4 - y^2*z^4 + 2*x*z^5 + 3*y*z^5 + 3*z^6 + 4/3*x^4*y - 2*x^3*y^2 + x^2*y^3 + x*y^4 - 3*y^5 - x^4*z - 2*x^2*y^2*z + 2*y^4*z + x^3*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 + 3*x^2*z^3 + 2*x*y*z^3 - 3*x*z^4 + y*z^4 - 1/2*x^3*y + 3*x^2*y^2 - 2*x*y^3 - y^4 + x^3*z + 2*x^2*y*z + 2*x*y^2*z - y^3*z + 3*x^2*z^2 - 2*x*y*z^2 + 2*x*z^3 + 3*y*z^3 + 3*z^4 - 3*x*y^2 + 3*y^3 - 3*y^2*z - 3*y*z^2 - 2*z^3', '3*x^8 - x^7*y + x^6*y^2 + 3*x^5*y^3 - 2*x^4*y^4 + 3*x^2*y^6 - 3*x*y^7 - y^8 + 2*x^7*z + 2*x^6*y*z - 3*x^5*y^2*z + x^4*y^3*z - 2*x^3*y^4*z + x^2*y^5*z + x*y^6*z + 3*y^7*z + 2*x^6*z^2 - 2*x^4*y^2*z^2 + x^3*y^3*z^2 + x^2*y^4*z^2 - 2*x*y^5*z^2 - y^6*z^2 + 2*x^4*y*z^3 - 2*x^3*y^2*z^3 - 2*x^2*y^3*z^3 - x*y^4*z^3 - 3*y^5*z^3 - 3*x^3*y*z^4 - 3*x^2*y^2*z^4 + 3*x*y^3*z^4 - 3*y^4*z^4 + 3*x^3*z^5 + x^2*y*z^5 - 2*x*y^2*z^5 + 2*y^3*z^5 + 3*x*y*z^6 + 3*y^2*z^6 - 3*x*z^7 + 3*y*z^7 + 3*z^8 - 1/840*x^7 - x^6*y + 2*x^4*y^3 + x^3*y^4 + x^2*y^5 - 3*x*y^6 - 3*y^7 + x^6*z - x^5*y*z - 2*x^4*y^2*z + 2*x^3*y^3*z + 2*x^2*y^4*z - 3*x*y^5*z - y^6*z - x^5*z^2 + 3*x^4*y*z^2 + 3*x^2*y^3*z^2 + 3*x*y^4*z^2 + 3*y^5*z^2 - x^4*z^3 + 3*x^3*y*z^3 - 3*x^2*y^2*z^3 + x*y^3*z^3 - 3*y^4*z^3 - x^3*z^4 + 3*x^2*y*z^4 - 2*x*y^2*z^4 + 2*y^3*z^4 + 2*x^2*z^5 - 3*x*y*z^5 + 2*y^2*z^5 + 3*x*z^6 - 3*y*z^6 + z^7 + x^4*y^2 + 3*x^2*y^4 + 3*x*y^5 + y^6 + 2*x^5*z + 3*x^4*y*z + 3*x^3*y^2*z - 2*x^2*y^3*z - 3*x*y^4*z - 3*y^5*z + 3*x^3*y*z^2 + 3*x^2*y^2*z^2 - 3*x*y^3*z^2 - 2*y^4*z^2 - 3*x^3*z^3 + 3*y^3*z^3 - 3*x^2*z^4 - 2*x*y*z^4 + 3*y^2*z^4 + 2*x*z^5 + y*z^5 - 3*z^6 + 13/12*x^4*y - x^3*y^2 - 3*x*y^4 - 2*y^5 - 3*x^4*z + 2*x^3*y*z + 2*x^2*y^2*z + 2*y^4*z + x^3*z^2 - x^2*y*z^2 + x*y^2*z^2 - 2*y^3*z^2 + 2*x^2*z^3 - 3*x*y*z^3 - 2*y^2*z^3 + 2*x*z^4 - y*z^4 - 7/12*x^3*y + 2*x*y^3 - 2*y^4 + x*y^2*z + y^3*z - 3/2*x^2*z^2 + 3*y^2*z^2 + 3*x*z^3 + y*z^3 - 3*z^4 - 2*x*y^2 + y^3 - 2*x*y*z - 3*y^2*z - z^3', 'x^8 + 3*x^7*y - 3*x^6*y^2 - x^4*y^4 + 2*x^3*y^5 - x^2*y^6 + 2*x*y^7 - 2*y^8 + 2*x^7*z - x^6*y*z + x^5*y^2*z + x^4*y^3*z - 2*x^3*y^4*z - 2*x^2*y^5*z - 2*x*y^6*z + 2*y^7*z + 2*x^6*z^2 + 3*x^3*y^3*z^2 - 2*x^2*y^4*z^2 + x*y^5*z^2 - 2*y^6*z^2 - x^5*z^3 + 3*x^4*y*z^3 + 3*x^2*y^3*z^3 + 2*x*y^4*z^3 - y^5*z^3 - 2*x^4*z^4 - 3*x^2*y^2*z^4 - 3*x*y^3*z^4 + 3*y^4*z^4 + 3*x^3*z^5 + 2*x^2*y*z^5 - 3*x*y^2*z^5 + y^3*z^5 + 2*x^2*z^6 - x*y*z^6 - 3*y^2*z^6 - x*z^7 + y*z^7 - 3*z^8 - 61/5040*x^7 + 2*x^6*y - 3*x^5*y^2 - 3*x^4*y^3 - x^3*y^4 + 2*x^5*y*z - 3*x^4*y^2*z - 3*x^3*y^3*z - 3*x^2*y^4*z - 2*x*y^5*z - y^6*z + 2*x^5*z^2 - x^4*y*z^2 - 2*x^3*y^2*z^2 + x^2*y^3*z^2 - 2*x*y^4*z^2 + 3*y^5*z^2 + 2*x^3*y*z^3 + x^2*y^2*z^3 - 3*x*y^3*z^3 - 3*y^4*z^3 + x^3*z^4 + 3*x^2*y*z^4 - 3*x*y^2*z^4 + 2*y^3*z^4 - 3*x^2*z^5 - 3*x*y*z^5 + 2*y^2*z^5 + x*z^6 - y*z^6 - z^7 - x^5*y + 3*x^4*y^2 - 2*x^2*y^4 - 2*x*y^5 + 2*y^6 + 2*x^4*y*z - 3*x^3*y^2*z - 2*x^2*y^3*z + 2*x*y^4*z + 3*y^5*z + 2*x^4*z^2 - x^2*y^2*z^2 + x*y^3*z^2 + 3*y^4*z^2 + 3*x^3*z^3 - 3*x^2*y*z^3 - 2*x*y^2*z^3 - y^3*z^3 - 2*x*y*z^4 - y^2*z^4 - 2*x*z^5 + y*z^5 + z^6 + 23/24*x^4*y - x^3*y^2 - x^2*y^3 + x*y^4 + 3*y^5 - 3*x^4*z - 3*x^2*y^2*z + 3*x*y^3*z - y^4*z + x^3*z^2 + x^2*y*z^2 - 3*x*y^2*z^2 - y^3*z^2 - 2*x^2*z^3 - 3*y^2*z^3 + x*z^4 + 3*y*z^4 - 2*z^5 + x^2*y^2 - 2*x*y^3 - y^4 - 3*x^3*z + 2*x^2*y*z - 2*x*y^2*z - y^3*z - 3*x^2*z^2 + 2*x*y*z^2 + 3*y^2*z^2 + x*z^3 + y*z^3 + 3*z^4 - 3*y^3 + 3*x*y*z - y*z^2 - 2*z^3']
  ok   R4 N1, component W4 (G_130 = P_203 = 0, linear): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.2 s
  ok   R4 N1, component W4 (G_130 = P_203 = 0, linear) anchor (A): fixed {'G_130': 0, 'P_203': 0}, solved ['G_211', 'G_103']; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 12)
      anchor generators: {'P': 'X^7 + 2*X^5*Y - Y^3 + X*Y*Z - 2*Y^2*Z - Z^3', 'G': 'X^4*Y + Y^3 + Y*Z^2 - Z^3', 'H': 'X^2*Z^2 + X*Y^2 + 2*Y^3 + 2*X*Y*Z + Y^2*Z - Y*Z^2'}
      anchor triple: ['3*x^6*y^2 + 2*x^4*y^4 + x^3*y^5 - x^2*y^6 - x*y^7 + 2*y^8 - x^6*y*z + x^5*y^2*z + x^4*y^3*z + 2*x^3*y^4*z + 2*x^2*y^5*z - x*y^6*z + 3*y^7*z + 2*x^6*z^2 + 2*x^5*y*z^2 + 3*x^4*y^2*z^2 + x^2*y^4*z^2 + 3*y^6*z^2 - x^5*z^3 + 2*x^4*y*z^3 - 2*x^3*y^2*z^3 + 2*x^2*y^3*z^3 + 2*x*y^4*z^3 + 2*x^4*z^4 + 3*x^3*y*z^4 + x^2*y^2*z^4 - 2*x*y^3*z^4 + 2*y^4*z^4 - 3*x^2*y*z^5 - 2*x*y^2*z^5 - 2*y^3*z^5 - 3*x^2*z^6 + 2*x*y*z^6 - 3*y^2*z^6 - x*z^7 + 2*y*z^7 - 2*z^8 + 1/20*x^7 + x^6*y + 2*x^5*y^2 + 2*x^4*y^3 + x^3*y^4 - 3*x^2*y^5 - 3*x*y^6 - 3*y^7 + x^5*y*z - 2*x^4*y^2*z - 3*x^3*y^3*z - x^2*y^4*z - x*y^5*z + 2*y^6*z - 2*x^5*z^2 - x^4*y*z^2 - x^3*y^2*z^2 + x^2*y^3*z^2 - 2*x*y^4*z^2 - y^5*z^2 - 2*x^4*z^3 + 2*x^3*y*z^3 + 3*x^2*y^2*z^3 - 2*x*y^3*z^3 + x^3*z^4 + x^2*y*z^4 + 2*x*y^2*z^4 - 3*y^3*z^4 - 3*x^2*z^5 + 3*x*y*z^5 - 2*y^2*z^5 + 3*x*z^6 + 2*y*z^6 - z^7 - 13/252*x^6 - x^5*y - 3*x^4*y^2 - x^3*y^3 - 2*x^2*y^4 - 2*x*y^5 + 3*x^5*z + 3*x^4*y*z - x^3*y^2*z + 2*x^2*y^3*z + 3*x*y^4*z - 2*y^5*z + 2*x^4*z^2 - 2*x^2*y^2*z^2 + 3*y^4*z^2 + 2*x^3*z^3 + 3*x^2*y*z^3 - x*y^2*z^3 - 2*y^3*z^3 + x^2*z^4 + 3*x*y*z^4 + x*z^5 + y*z^5 - z^6 + 13/12*x^4*y - 2*x^3*y^2 - 3*x^2*y^3 + 3*x*y^4 - y^5 - 2*x^4*z - 3*x^3*y*z - 2*x^2*y^2*z - x*y^3*z + 2*y^4*z - x^2*y*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 + 3*x^2*z^3 + x*y*z^3 + x*z^4 + y*z^4 + 2*x^2*y^2 + 2*x*y^3 + 2*y^4 - 2*x^2*y*z + 3*x*y^2*z - 2*y^3*z + 11/2*x^2*z^2 + 3*x*y*z^2 + 2*y^2*z^2 - y*z^3 - 2*z^4 - 3*x*y^2 - 2*y^3 + 3*y^2*z - y*z^2 + 2*z^3', '-3*x^8 + 2*x^7*y + x^6*y^2 + 2*x^5*y^3 - x^4*y^4 + 2*x^2*y^6 - 3*y^8 - x^7*z - x^5*y^2*z + 2*x^4*y^3*z + 2*x^3*y^4*z - 3*x^2*y^5*z - 2*x*y^6*z - 2*y^7*z + 2*x^6*z^2 + 2*x^4*y^2*z^2 - x^3*y^3*z^2 - x^2*y^4*z^2 - x*y^5*z^2 - 2*y^6*z^2 + 2*x^5*z^3 - 3*x^4*y*z^3 - x^2*y^3*z^3 + x*y^4*z^3 - 2*y^5*z^3 + x^4*z^4 - 2*x^2*y^2*z^4 - 2*x*y^3*z^4 + 2*y^4*z^4 + 3*x^3*z^5 - x^2*y*z^5 + x*y^2*z^5 - 3*x^2*z^6 - 2*x*y*z^6 + 2*x*z^7 - 3*y*z^7 - z^8 - 79/560*x^7 + 3*x^6*y - 3*x^5*y^2 - x^4*y^3 + 3*x^3*y^4 + 2*x^2*y^5 + 2*x*y^6 - 2*y^7 - 3*x^6*z + x^5*y*z + x^4*y^2*z + 3*x^2*y^4*z - 3*x*y^5*z + 3*y^6*z - 2*x^5*z^2 - 2*x^4*y*z^2 - 3*x^3*y^2*z^2 - 3*x^2*y^3*z^2 + x*y^4*z^2 + y^5*z^2 - x^4*z^3 - x^3*y*z^3 - 3*x^2*y^2*z^3 + 2*x*y^3*z^3 + 2*y^4*z^3 - x^3*z^4 - x^2*y*z^4 - 2*x*y^2*z^4 + y^3*z^4 + 2*x*y*z^5 + 2*y^2*z^5 + 2*x*z^6 - 2*y*z^6 - 5/84*x^6 + 3*x^5*y - 3*x^4*y^2 + 3*x^3*y^3 - x*y^5 - 3*y^6 - x^5*z - 3*x^4*y*z - 2*x^3*y^2*z + 3*x^2*y^3*z - 3*x*y^4*z + y^5*z + x^4*z^2 - 2*x^3*y*z^2 - x^2*y^2*z^2 + 2*x*y^3*z^2 - 2*y^4*z^2 + 3*x^3*z^3 + 3*x^2*y*z^3 - 2*x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 3*x*y*z^4 - 2*y^2*z^4 - 2*x*z^5 + 2*y*z^5 - 3*z^6 + 5/4*x^4*y - x^3*y^2 - x^2*y^3 - x*y^4 - 3*y^5 - 3*x^4*z - 2*x^3*y*z - x^2*y^2*z + 3*y^4*z + 3*x^3*z^2 + 3*x^2*y*z^2 - 3*x*y^2*z^2 - y^3*z^2 - 2*x^2*z^3 + 3*x*y*z^3 - y^2*z^3 + y*z^4 + z^5 - 2*x^2*y^2 + 2*y^4 + 2*x^3*z + 2*x^2*y*z - 2*x*y^2*z + y^3*z + 23/2*x^2*z^2 + 3*x*y*z^2 + 3*y^2*z^2 - 2*x*z^3 - y*z^3 + 2*z^4 - 3*x*y^2 - 3*y^3 - x*y*z + 2*y^2*z + 3*y*z^2 + 3*z^3', '2*x^8 + 3*x^7*y + x^6*y^2 - x^3*y^5 + 2*x^2*y^6 - 2*x*y^7 - 2*y^8 - x^6*y*z - 2*x^5*y^2*z - 2*x^4*y^3*z + x^3*y^4*z + 2*x^2*y^5*z - y^7*z - 3*x^4*y^2*z^2 - 3*x^3*y^3*z^2 + x*y^5*z^2 - y^6*z^2 + 2*x^5*z^3 + 2*x^4*y*z^3 + x^3*y^2*z^3 + 3*x^2*y^3*z^3 + 3*x*y^4*z^3 - 3*y^5*z^3 - x^3*y*z^4 + y^4*z^4 - 3*x^2*y*z^5 + x*y^2*z^5 + 2*y^3*z^5 + 3*x^2*z^6 + 2*x*y*z^6 + 3*x*z^7 + 3*y*z^7 + 2*z^8 + 163/1680*x^7 - 3*x^6*y - 2*x^5*y^2 - x^4*y^3 - x^3*y^4 - x^2*y^5 - x*y^6 - 3*y^7 + 2*x^5*y*z - x^4*y^2*z + 2*x^3*y^3*z + x^2*y^4*z - 2*x*y^5*z + 2*x^5*z^2 + 3*x^4*y*z^2 - 3*x^3*y^2*z^2 + x*y^4*z^2 - y^5*z^2 - 3*x^4*z^3 + 2*x^3*y*z^3 + x^2*y^2*z^3 + 2*y^4*z^3 - 2*x^3*z^4 - 3*x^2*y*z^4 + x*y^2*z^4 + 3*y^3*z^4 - 2*x^2*z^5 - x*y*z^5 - 3*y^2*z^5 - 3*y*z^6 + z^7 + 1/28*x^6 - 2*x^5*y + 2*x^3*y^3 + 3*x^2*y^4 + x*y^5 + 3*y^6 - 3*x^5*z + x^3*y^2*z - 3*x^2*y^3*z - 2*y^5*z + 2*x^4*z^2 - x^3*y*z^2 - 3*x^2*y^2*z^2 + 3*x*y^3*z^2 + 3*y^4*z^2 + 3*x^3*z^3 - 2*x^2*y*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 + 2*x*y*z^4 - 3*y^2*z^4 - x*z^5 + 3*y*z^5 - z^6 - 3/4*x^4*y - 3*x^3*y^2 + 2*x^2*y^3 - 3*x*y^4 - 2*y^5 + 2*x^3*y*z - 2*x^2*y^2*z + x*y^3*z + 3*y^4*z + x^3*z^2 - 3*x^2*y*z^2 + x*y^2*z^2 - 3*y^3*z^2 + 2*x^2*z^3 - 2*x*y*z^3 - 2*y^2*z^3 + y*z^4 + 2*z^5 - 2*x^2*y^2 - 2*x*y^3 - 3*y^4 + 2*x^3*z - 2*x^2*y*z - 3*x*y^2*z - y^3*z - 3*x^2*z^2 - 3*y^2*z^2 - 2*x*z^3 - 3*y*z^3 - 2*z^4 - 3*x*y^2 + y^3 + 3*x*y*z + 3*y^2*z + 3*y*z^2 - z^3']
--- R4 N5
  ok   R4 N5: the graded normal form is closed with H = [1, 3, 6, 4, 3, 2, 1, 1]
  ok   R4 N5: one lift per pivot of D_4..D_7
    134 closedness conditions in 88 tail coefficients; 55 eliminated with constant pivots; residual equations:
       (-15/4) * G_130 * (2*P_501 - 7*G_211)
       (105) * G_130 * G_202
       (315/2) * G_103 * G_130
       (1/84) * (20*P_510*P_501*G_220 - 10*P_510^2*G_211 + 40*P_501^2*H_120 - 20*P_510*P_501*H_111 - 12*P_501*P_320 + 6*P_510*P_311 - 42*P_311*G_220 + 42*P_320*G_211 - 252*P_401*G_130 + 2205*G_130*G_111 - 168*P_302*H_120 + 42*P_311*H_111)
       (1/252) * (-210*P_501*G_220*G_130 + 105*P_510*G_211*G_130 + 120*P_501^2*H_030 - 40*P_510*P_501*H_021 - 63*P_311*G_130 + 4410*G_130*G_021 - 504*P_302*H_030 + 84*P_311*H_021)
       (1/84) * (-105*P_501*G_211*G_130 + 210*P_510*G_202*G_130 + 40*P_501^2*H_021 - 40*P_510*P_501*H_012 + 4410*G_130*G_012 - 168*P_302*H_021 + 84*P_311*H_012)
       (1/42) * (-105*P_501*G_202*G_130 - 15*P_510^2*G_103 + 20*P_501^2*H_012 - 60*P_510*P_501*H_003 + 63*P_320*G_103 + 6615*G_130*G_003 - 84*P_302*H_012 + 126*P_311*H_003)
       (-15/14) * G_130 * (2*P_501 - 7*G_211)
       (30) * G_130 * G_202
       (45/2) * G_103 * G_130
  ok   R4 N5: on {G_130 != 0} the residual equations are triangular in ['G_211', 'G_202', 'G_103', 'G_111', 'G_021', 'G_012', 'G_003'] with diagonal (const)*G_130: a graph (irreducible)
  ok   R4 N5: on {G_130 = 0}: {5*P_501^2 - 21*P_302 != 0}: triangular in ['H_012', 'H_021', 'H_030', 'H_120']; {-10*P_510*P_501 + 21*P_311 != 0}: triangular in ['H_021', 'H_012', 'H_003', 'H_111']; then c*(linear forms), c = -5*P_510^2 + 21*P_320: ['-1/147*P_501 + 1/42*G_211', '1/14*G_103']
  ok   R4 N5, component W1 (G_130 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.3 s, stabilization 0.1 s
  ok   R4 N5, component W1 (G_130 != 0) anchor (A): fixed {'G_130': 1}, solved ['G_211', 'G_202', 'G_103', 'G_111', 'G_021', 'G_012', 'G_003']; member closed with H and the prescribed graded data: True; a triple of I of local length 35 (ds), stabilization (35, 10)
      anchor generators: {'P': 'X^7 + 105/4*X^4*Y^2 + 105/8*X*Y^4 - 35/2*X*Y^3 + 105/4*Y^4 - X*Y^2 + 2*Y^3 - 2*X*Y*Z + 2*Y^2*Z - 2*Y*Z^2 - 2*Z^3', 'G': 'X^4*Y + X*Y^3 - X*Y^2 + 2*Y^3', 'H': 'X^3*Z - X*Y^2 - Y^3 + 2*X*Y*Z - 2*Y^2*Z + 2*Z^3'}
      anchor triple: ['3*x^7*y + 3*x^4*y^4 - x^3*y^5 - 3*x*y^7 + 2*y^8 + 2*x^7*z + x^6*y*z - 2*x^5*y^2*z - 2*x^4*y^3*z - x^3*y^4*z + x^2*y^5*z + 2*x*y^6*z + 3*x^5*y*z^2 + 3*x^4*y^2*z^2 - 2*x^3*y^3*z^2 + 2*x^2*y^4*z^2 - x*y^5*z^2 - y^6*z^2 + x^5*z^3 + x^4*y*z^3 - 2*x^3*y^2*z^3 + x^2*y^3*z^3 - 2*x*y^4*z^3 - 2*x^4*z^4 + x^2*y^2*z^4 + 3*x*y^3*z^4 + y^4*z^4 + 2*x^3*z^5 - 3*x^2*y*z^5 - 3*y^3*z^5 - 3*x^2*z^6 + 2*x*y*z^6 + y^2*z^6 + 3*x*z^7 + 3*z^8 - 383/840*x^7 + x^6*y - x^5*y^2 + 2*x^4*y^3 + 2*x^3*y^4 + 2*x^2*y^5 + x*y^6 + 3*y^7 + x^6*z - x^5*y*z - x^4*y^2*z - 2*y^6*z + 2*x^5*z^2 - 3*x^4*y*z^2 + x^3*y^2*z^2 - 3*x^2*y^3*z^2 + x*y^4*z^2 - y^5*z^2 - 3*x^4*z^3 - 3*x^3*y*z^3 + 3*x^2*y^2*z^3 - 3*x*y^3*z^3 + y^4*z^3 - 3*x^3*z^4 + 2*x^2*y*z^4 + x*y^2*z^4 - 2*y^3*z^4 + x^2*z^5 - 2*x*y*z^5 - 2*y^2*z^5 - 3*x*z^6 + 2*y*z^6 + 1/8*x^6 + 2*x^5*y + 2*x^4*y^2 + 2*x^2*y^4 + 2*x*y^5 + y^6 + 3*x^5*z - 2*x^4*y*z + 2*x^3*y^2*z - 2*x^2*y^3*z - 3*x*y^4*z + 2*y^5*z - x^4*z^2 + 2*x^2*y^2*z^2 + 3*x*y^3*z^2 - 2*y^4*z^2 - 3*x^3*z^3 - x^2*y*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x^2*z^4 - 3*x*y*z^4 + 2*y^2*z^4 + x*z^5 + 2*y*z^5 - 3*z^6 - 1/2*x^5 + x^4*y - x^3*y^2 + 3*x^2*y^3 - 2*x*y^4 - y^5 - 2*x^4*z - x^3*y*z + 2*x^2*y^2*z - x*y^3*z - 2*y^4*z + 2*x^3*z^2 + 2*x^2*y*z^2 + 2*x*y^2*z^2 - 3*y^3*z^2 + 3*x^2*z^3 - 2*x*y*z^3 + 3*y^2*z^3 - 3*y*z^4 - 3*z^5 + 3/4*x^3*y + 2*x^2*y^2 + 2*x*y^3 + y^4 + x^3*z + 3*x^2*y*z + 3*x*y^2*z + 2*y^3*z + x*y*z^2 + y^2*z^2 - 2*x*z^3 + 2*y*z^3 + 3*z^4 - 3*y^3 - 3*y^2*z - 3*z^3', '-x^8 - 3*x^7*y + 3*x^6*y^2 - x^5*y^3 - 3*x^4*y^4 - x^3*y^5 - 2*x*y^7 + y^8 + x^7*z + 3*x^6*y*z - 3*x^3*y^4*z + x^2*y^5*z - 2*x*y^6*z + 2*y^7*z - 2*x^6*z^2 + 3*x^5*y*z^2 + 2*x^4*y^2*z^2 - 2*x^2*y^4*z^2 + 3*x*y^5*z^2 - 3*y^6*z^2 + 3*x^5*z^3 + 2*x^4*y*z^3 + 2*x^3*y^2*z^3 - 2*x^2*y^3*z^3 + x*y^4*z^3 - 2*x^4*z^4 + 3*x^2*y^2*z^4 + 2*y^4*z^4 + 2*x^3*z^5 + x^2*y*z^5 + 2*y^3*z^5 + 2*x^2*z^6 - 2*x*y*z^6 - y^2*z^6 + x*z^7 + 2*y*z^7 - 3*z^8 + 3937/5040*x^7 - 3*x^6*y + x^5*y^2 - x^4*y^3 + x^3*y^4 + 2*x^2*y^5 + x*y^6 - 3*y^7 + x^6*z + 2*x^5*y*z + x^4*y^2*z + 3*x^3*y^3*z - x^2*y^4*z - x*y^5*z + 2*y^6*z - 2*x^5*z^2 + 3*x^4*y*z^2 - 2*x^2*y^3*z^2 + y^5*z^2 - x^4*z^3 - 2*x^3*y*z^3 + 3*x^3*z^4 - 2*x^2*y*z^4 - 3*x*y^2*z^4 + 2*y^3*z^4 - 2*x^2*z^5 + 2*y^2*z^5 - x*z^6 - 3*y*z^6 - 2*z^7 + 5/6*x^6 + x^5*y - 3*x^4*y^2 + 2*x^3*y^3 + x^2*y^4 - 3*x*y^5 + 3*y^6 - x^5*z - 2*x^4*y*z - x^3*y^2*z + 3*x^2*y^3*z + 3*x*y^4*z + 2*y^5*z + x^4*z^2 - x^3*y*z^2 + 2*x^2*y^2*z^2 + x*y^3*z^2 + 3*y^4*z^2 - 2*x^2*y*z^3 - 2*x*y^2*z^3 - 3*y^3*z^3 - 2*x^2*z^4 + x*y*z^4 + 2*y^2*z^4 + 2*x*z^5 - 3*y*z^5 + 3*z^6 + 1/4*x^5 + 5/4*x^4*y - 3*x^3*y^2 - 2*x^2*y^3 + 3*x*y^4 - 2*y^5 + 2*x^4*z - x^3*y*z + 2*x^2*y^2*z - 3*x*y^3*z + 3*y^4*z + 3*x^3*z^2 - x^2*y*z^2 + y^3*z^2 - x^2*z^3 - 3*x*y*z^3 + 2*y^2*z^3 + 3*x*z^4 + 3*y*z^4 + 3*z^5 + 3/4*x^4 + 1/2*x^3*y - x^2*y^2 - 2*x*y^3 - 2*y^4 - 8*x^3*z + x^2*y*z + 3*x*y^2*z - x^2*z^2 - x*y*z^2 - 3*y^2*z^2 + 2*y*z^3 - 3*x*y^2 - 2*y^3 - x*y*z + y^2*z + y*z^2 + 3*z^3', '-x^8 - x^7*y + x^5*y^3 + 2*x^4*y^4 - 2*x^3*y^5 + 3*x^2*y^6 + 2*x*y^7 - y^8 + x^7*z - 2*x^6*y*z - 3*x^5*y^2*z - 3*x^4*y^3*z - 2*x^3*y^4*z - x^2*y^5*z - 3*x*y^6*z - 3*y^7*z - x^6*z^2 - 2*x^5*y*z^2 - 2*x^3*y^3*z^2 - 2*x^2*y^4*z^2 + 2*y^6*z^2 + 3*x^5*z^3 - 2*x^4*y*z^3 - x^3*y^2*z^3 + 2*x^2*y^3*z^3 - 2*x*y^4*z^3 - 3*y^5*z^3 + x^4*z^4 + 2*x^3*y*z^4 + 3*x*y^3*z^4 + 3*y^4*z^4 + 3*x^3*z^5 + y^3*z^5 + 2*x^2*z^6 + 2*x*y*z^6 - 3*y^2*z^6 - 2*x*z^7 + z^8 - 1/24*x^7 - 2*x^6*y - 2*x^5*y^2 + 2*x^4*y^3 + x^3*y^4 - 3*x^2*y^5 + 2*x*y^6 - 3*y^7 - 2*x^6*z + 2*x^5*y*z + 3*x^4*y^2*z - 3*x^3*y^3*z + 3*x*y^5*z + y^6*z + x^5*z^2 + x^4*y*z^2 + x^3*y^2*z^2 - 3*x^2*y^3*z^2 + 2*x*y^4*z^2 - 2*y^5*z^2 - 3*x^4*z^3 - x^3*y*z^3 - 3*x^2*y^2*z^3 + x*y^3*z^3 + 2*y^4*z^3 + 2*x^3*z^4 + x^2*y*z^4 - x*y^2*z^4 - 3*y^3*z^4 - 2*x^2*z^5 - x*y*z^5 + 3*y^2*z^5 - 2*x*z^6 - z^7 - 7/16*x^6 + 3*x^5*y + x^4*y^2 - 3*x^3*y^3 + 2*x^2*y^4 + 2*x*y^5 - y^6 - 3*x^5*z - x^4*y*z + 2*x^3*y^2*z - x^2*y^3*z - 3*x*y^4*z + 2*y^5*z + 2*x^3*y*z^2 + 3*x^2*y^2*z^2 + 2*x*y^3*z^2 + 3*y^4*z^2 - 3*x^3*z^3 - 3*x^2*y*z^3 + 2*x*y^2*z^3 - 3*y^3*z^3 - 3*x^2*z^4 + x*z^5 + 2*y*z^5 - 1/2*x^5 + 1/4*x^4*y + 2*x^3*y^2 - 3*x^2*y^3 - 2*x*y^4 + y^5 - 3*x^4*z + x^3*y*z + x^2*y^2*z + y^4*z + x^3*z^2 + x^2*y*z^2 - 2*x*y^2*z^2 - x*y*z^3 + y^2*z^3 - 3*x*z^4 - y*z^4 - z^5 + 3/4*x^4 + 2*x^2*y^2 - 2*x*y^3 - y^4 - 4*x^3*z - x^2*y*z + 3*x*y^2*z + y^3*z + 3*y^2*z^2 + x*z^3 + 2*y*z^3 - 3*x*y^2 + x*y*z + 2*y^2*z - 3*y*z^2 + 2*z^3']
  ok   R4 N5, component W2 (G_130 = 0, beta != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.2 s
  ok   R4 N5, component W2 (G_130 = 0, beta != 0) anchor (A): fixed {'G_130': 0}, solved ['H_012', 'H_021', 'H_030', 'H_120']; member closed with H and the prescribed graded data: True; a triple of I of local length 36 (ds), stabilization (36, 11)
      anchor generators: {'P': 'X^7 + X^5*Y - 2*X^5*Z + 2*X^3*Y^2 - 2*X^3*Y*Z - 2*X^3*Z^2 + 296/1953*X*Y^3 + 14641/5004888*Y^4 + 8/21*X*Y^2*Z + 1331/20181*Y^3*Z + 68/21*X*Y*Z^2 + 121/217*Y^2*Z^2 + 44/21*Y*Z^3 + 62/21*Z^4 - X*Y^2 - 2*Y^3 - 2*X*Y*Z + 2*Y^2*Z + 2*Y*Z^2 + Z^3', 'G': 'X^4*Y + 2*Y^3 - X*Y*Z - 2*Y^2*Z', 'H': 'X^3*Z - 5/31*X*Y^2 - 1331/119164*Y^3 - X*Y*Z - 363/1922*Y^2*Z - 33/31*Y*Z^2 - 2*Z^3'}
      anchor triple: ['x^8 - x^7*y + 2*x^6*y^2 + x^5*y^3 - 2*x^4*y^4 + x^3*y^5 - 3*x^2*y^6 - 2*x*y^7 - 2*y^8 + x^7*z + x^6*y*z + 3*x^5*y^2*z - x^4*y^3*z + x^3*y^4*z + x*y^6*z + 3*y^7*z + 3*x^6*z^2 - 3*x^5*y*z^2 - x^3*y^3*z^2 + x^2*y^4*z^2 - 2*x*y^5*z^2 + 2*y^6*z^2 - 2*x^5*z^3 - 2*x^4*y*z^3 + x^3*y^2*z^3 + x^2*y^3*z^3 - y^5*z^3 - 2*x^4*z^4 + 3*x^3*y*z^4 - 2*x*y^3*z^4 - 2*x^3*z^5 - 3*x^2*y*z^5 - 2*x*y^2*z^5 + y^3*z^5 - 3*x^2*z^6 + 3*x*y*z^6 + y^2*z^6 - x*z^7 + 2*y*z^7 - z^8 + 30583607/394134930*x^7 + 2*x^6*y - 2*x^5*y^2 + x^4*y^3 - 2*x^3*y^4 + 3*x^2*y^5 + 2*x*y^6 - 3*y^7 - 2*x^5*y*z + x^4*y^2*z - 3*x^3*y^3*z + 2*x^2*y^4*z + x*y^5*z + 3*y^6*z - x^5*z^2 - x^4*y*z^2 + x^3*y^2*z^2 - 3*x*y^4*z^2 - 2*y^5*z^2 - 3*x^4*z^3 - 2*x^3*y*z^3 + x^2*y^2*z^3 - 2*x*y^3*z^3 + 3*y^4*z^3 - x^3*z^4 - 2*x^2*y*z^4 - 3*x*y^2*z^4 - 3*x^2*z^5 - 2*x*y*z^5 + y^2*z^5 + x*z^6 + 3*y*z^6 + 2*z^7 + 25747/1640520*x^6 - x^5*y - x^4*y^2 + 3*x^3*y^3 - 3*x^2*y^4 - 3*x*y^5 - 3*y^6 + x^5*z - 2*x^4*y*z - 3*x^3*y^2*z + x^2*y^3*z + 3*y^5*z - x^4*z^2 + 3*x^3*y*z^2 + 2*x^2*y^2*z^2 - 3*x*y^3*z^2 - 3*y^4*z^2 + x^3*z^3 + 3*x^2*y*z^3 - x*y^2*z^3 + 3*y^3*z^3 - 3*x^2*z^4 + 3*x*z^5 + y*z^5 + 2*z^6 + 7359833/37536660*x^5 + 13/12*x^4*y + x^3*y^2 - 2*x^2*y^3 + 3*x*y^4 + 2*y^5 + x^4*z - x^3*y*z - 3*x^2*y^2*z + x*y^3*z + 3*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x*y*z^3 - 2*y^2*z^3 + 3*x*z^4 - 3*y*z^4 - 2*z^5 + 3*x^2*y^2 - 2*y^4 + 1543465/357492*x^3*z - x^2*y*z + 2*x*y^2*z + 2*y^3*z + 2*x^2*z^2 - 3*y^2*z^2 - 2*x*z^3 - y*z^3 + 3*x*y^2 - y^3 + 2*x*y*z + 3*y^2*z - y*z^2 + 2*z^3', '-2*x^8 + 2*x^7*y + 2*x^6*y^2 - 3*x^5*y^3 - x^4*y^4 - 3*x^3*y^5 + 3*x^2*y^6 + 3*x*y^7 - 3*y^8 + 2*x^7*z + x^6*y*z + 3*x^5*y^2*z + x^3*y^4*z - x^2*y^5*z - 2*x*y^6*z + y^7*z + 3*x^6*z^2 + 3*x^5*y*z^2 + 3*x^4*y^2*z^2 + 2*x^3*y^3*z^2 - 2*x*y^5*z^2 + 3*y^6*z^2 + x^5*z^3 + 2*x^4*y*z^3 - 2*x^3*y^2*z^3 + x^2*y^3*z^3 - y^5*z^3 - 3*x^4*z^4 + x^2*y^2*z^4 - 3*x*y^3*z^4 - 3*y^4*z^4 - x^3*z^5 - 2*x^2*y*z^5 - 3*x*y^2*z^5 - 2*y^3*z^5 + x^2*z^6 - x*y*z^6 - y^2*z^6 - 3*x*z^7 + 2*y*z^7 + 2*z^8 - 2853751/197067465*x^7 - 2*x^6*y + 3*x^5*y^2 + x^4*y^3 + x^3*y^4 - 2*x^2*y^5 - 2*x*y^6 - y^7 - 3*x^6*z - 3*x^5*y*z + 3*x^4*y^2*z - 2*x^3*y^3*z - 3*x^2*y^4*z - 3*x*y^5*z - 2*y^6*z + 2*x^5*z^2 - 2*x^4*y*z^2 + 2*x^3*y^2*z^2 - 3*x*y^4*z^2 + 2*y^5*z^2 - x^3*y*z^3 + x^2*y^2*z^3 - y^4*z^3 + 2*x^3*z^4 - x^2*y*z^4 + 3*x*y^2*z^4 + 3*y^3*z^4 - x^2*z^5 + x*y*z^5 + 3*y^2*z^5 - x*z^6 + y*z^6 - 2*z^7 - 69439/364560*x^6 - 2*x^5*y + x^4*y^2 - x^3*y^3 - x^2*y^4 - 2*x*y^5 + y^6 - x^5*z + 2*x^4*y*z + x^3*y^2*z + 3*x^2*y^3*z - 3*x*y^4*z + y^5*z + 2*x^4*z^2 + x^3*y*z^2 - 2*x^2*y^2*z^2 - 3*x*y^3*z^2 + 2*x^3*z^3 - 3*x^2*y*z^3 + 2*x*y^2*z^3 + y^3*z^3 + 3*y*z^5 - 170119/3753666*x^5 + 41/24*x^4*y + x^3*y^2 - x^2*y^3 + 3*x*y^4 - y^5 - 3*x^4*z + 2*x^3*y*z - 3*x^2*y^2*z + 2*x*y^3*z + 3*y^4*z - x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 + 2*y^3*z^2 + 3*x^2*z^3 + 2*x*y*z^3 + 2*y^2*z^3 + 3*x*z^4 + y*z^4 + 2*z^5 + x*y^3 + 2*y^4 - 429611/357492*x^3*z + 2*x*y^2*z + y^3*z - x^2*z^2 + x*y*z^2 - 2*y^2*z^2 - 2*x*y^2 - 3*y^3 + x*y*z + y^2*z + 2*y*z^2 - z^3', 'x^8 + 3*x^7*y + 3*x^6*y^2 + x^5*y^3 - x^4*y^4 - 3*x^3*y^5 + 3*x^2*y^6 - 3*x*y^7 - y^8 - 2*x^7*z + 3*x^6*y*z + 3*x^4*y^3*z + 3*x^3*y^4*z + 2*x^2*y^5*z - 3*x*y^6*z + 2*y^7*z + 2*x^6*z^2 - 3*x^4*y^2*z^2 - x^2*y^4*z^2 + x*y^5*z^2 - 3*y^6*z^2 - 2*x^3*y^2*z^3 + 2*x^2*y^3*z^3 - 3*y^5*z^3 + 3*x^4*z^4 + 2*x^3*y*z^4 - x^2*y^2*z^4 - x*y^3*z^4 - x^3*z^5 - 3*x^2*y*z^5 - x*y^2*z^5 - y^3*z^5 + 3*x^2*z^6 + 3*x*y*z^6 - y^2*z^6 + 3*x*z^7 + 3*y*z^7 - 3*z^8 + 616967723/3153079440*x^7 - x^6*y + 2*x^5*y^2 + 3*x^3*y^4 - x^2*y^5 + x*y^6 - 3*y^7 + x^6*z - x^5*y*z - 3*x^3*y^3*z - 2*x^2*y^4*z + 2*x*y^5*z - 3*y^6*z - 2*x^5*z^2 + 3*x^3*y^2*z^2 - x^2*y^3*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 - 2*x^3*y*z^3 + x*y^3*z^3 + 3*y^4*z^3 + 2*x^3*z^4 + x*y^2*z^4 - 2*y^3*z^4 + 2*x^2*z^5 + 2*x*y*z^5 - 2*y^2*z^5 - 3*x*z^6 - 3*y*z^6 - z^7 - 48487/3281040*x^6 - 2*x^5*y - 3*x^4*y^2 + x^3*y^3 - 2*x*y^5 + 2*x^5*z - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z + 2*x*y^4*z - 2*y^5*z - x^4*z^2 + 3*x^3*y*z^2 - 2*x^2*y^2*z^2 - 3*x*y^3*z^2 + y^4*z^2 - 3*x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + 3*x^2*z^4 + x*y*z^4 - 3*y^2*z^4 + 3*x*z^5 + 3*y*z^5 - z^6 - 480022/3128055*x^5 - 7/24*x^4*y - 3*x^3*y^2 - x*y^4 - y^5 - x^4*z + 3*x^2*y^2*z - x*y^3*z - 3*x^3*z^2 - 3*x^2*y*z^2 + 3*x*y^2*z^2 + 2*y^3*z^2 + 3*x^2*z^3 + 3*x*y*z^3 + y^2*z^3 + x*z^4 + 3*y*z^4 + 3*z^5 - 3*x^2*y^2 - 3*y^4 - 366143/119164*x^3*z + 2*x^2*y*z - 3*x*y^2*z - 3*y^3*z + x^2*z^2 + x*y*z^2 - 3*y^2*z^2 + 2*x*z^3 - 3*y*z^3 - 3*z^4 + y^3 - 3*x*y*z + 2*y^2*z - 2*y*z^2 - z^3']
  ok   R4 N5, component W3 (G_130 = 0 = beta, gamma != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.3 s, stabilization 0.1 s
  ok   R4 N5, component W3 (G_130 = 0 = beta, gamma != 0) anchor (A): fixed {'G_130': 0, 'P_501': 0, 'P_302': 0}, solved ['H_021', 'H_012', 'H_003', 'H_111']; member closed with H and the prescribed graded data: True; a triple of I of local length 36 (ds), stabilization (36, 10)
      anchor generators: {'P': 'X^7 + X^3*Y^2 + 2*X^3*Y*Z + 2/3*X*Y^3 - 1/2*Y^4 - 2*X*Y^2 + 2*Y^3 + X*Y*Z + Y^2*Z + 2*Y*Z^2 - 2*Z^3', 'G': 'X^4*Y - X*Y^2 - 2*X*Y*Z + 2*Y*Z^2 + Z^3', 'H': 'X^3*Z + X*Y^2 - Y^3'}
      anchor triple: ['2*x^8 - x^7*y + 2*x^6*y^2 - 2*x^5*y^3 - 2*x^4*y^4 - 2*x^3*y^5 + 2*x^2*y^6 + 2*y^8 + 3*x^7*z - 2*x^6*y*z + 2*x^5*y^2*z - 2*x^4*y^3*z + 3*x^3*y^4*z - x^2*y^5*z - 3*x*y^6*z + x^6*z^2 + x^5*y*z^2 + 2*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + x^2*y^4*z^2 + 3*x*y^5*z^2 - y^6*z^2 - 3*x^5*z^3 + x^4*y*z^3 - x^2*y^3*z^3 - 2*x*y^4*z^3 + 2*x^4*z^4 + 2*x^3*y*z^4 + 2*x^2*y^2*z^4 + 2*x*y^3*z^4 - 3*y^4*z^4 - 2*x^2*y*z^5 - 3*x*y^2*z^5 + 2*y^3*z^5 + 3*x^2*z^6 - 3*x*y*z^6 - y^2*z^6 + 2*x*z^7 - 3*y*z^7 + z^8 - 1/90*x^7 + 2*x^5*y^2 + 2*x^4*y^3 - 2*x^3*y^4 + 3*x^2*y^5 - y^7 + 2*x^6*z + 3*x^5*y*z - x^4*y^2*z - x^3*y^3*z - 3*x^2*y^4*z + x*y^5*z + 2*y^6*z - 3*x^5*z^2 - 2*y^5*z^2 - 2*x^4*z^3 + x*y^3*z^3 + 3*y^4*z^3 + 2*x^3*z^4 - x^2*y*z^4 - y^3*z^4 - 3*x^2*z^5 + x*y*z^5 - 3*y^2*z^5 + 3*x*z^6 - 3*z^7 + 1/252*x^6 + x^5*y + x^4*y^2 - x^3*y^3 - 2*x^2*y^4 - 3*x*y^5 + y^6 + x^5*z + 3*x^4*y*z + x^3*y^2*z - 3*x^2*y^3*z - y^5*z + 3*x^4*z^2 - 2*x^3*y*z^2 - 2*x^2*y^2*z^2 - 3*x*y^3*z^2 + 3*y^4*z^2 - 3*x^3*z^3 + x*y^2*z^3 - 2*y^3*z^3 - 3*x^2*z^4 - 2*x*y*z^4 - 2*y^2*z^4 - y*z^5 + 2*z^6 - 1/140*x^5 + 7/6*x^4*y + 3*x^3*y^2 + 3*x^2*y^3 + y^5 - 2*x^4*z - 2*x^3*y*z + 2*x^2*y^2*z + 2*x*y^3*z - 3*y^4*z - 3*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 + 2*y^3*z^2 - 3*x^2*z^3 + 3*y^2*z^3 - 3*x*z^4 + 2*y*z^4 + x^2*y^2 + 2*x*y^3 - 3*x^2*y*z + 2*x*y^2*z - 3*x^2*z^2 - x*y*z^2 - 2*y^2*z^2 - x*z^3 + y*z^3 + 2*z^4 + 3*x*y^2 + y^3 + 2*y^2*z - y*z^2 - 3*z^3', '-3*x^8 + x^7*y + 2*x^4*y^4 + 3*x^2*y^6 + 2*x*y^7 - 2*y^8 - x^7*z + x^6*y*z + 3*x^5*y^2*z + 2*x^4*y^3*z - x^2*y^5*z + 3*y^7*z + 3*x^6*z^2 + 2*x^4*y^2*z^2 + 3*x^3*y^3*z^2 + x^2*y^4*z^2 - 3*y^6*z^2 + 2*x^5*z^3 + 3*x^4*y*z^3 - x^3*y^2*z^3 - 2*x*y^4*z^3 + 2*y^5*z^3 - x^4*z^4 - x^3*y*z^4 - 2*x^2*y^2*z^4 - 2*x*y^3*z^4 - 2*y^4*z^4 + 3*x^3*z^5 - 2*x^2*y*z^5 - x*y^2*z^5 - 3*y^3*z^5 - x^2*z^6 + 3*x*y*z^6 - y^2*z^6 + x*z^7 - y*z^7 + 3*z^8 + 1/112*x^7 - x^6*y + 3*x^5*y^2 - x^4*y^3 - 3*x^3*y^4 - 2*x^2*y^5 - 3*y^7 - 3*x^6*z - 3*x^5*y*z - 2*x^4*y^2*z + 3*x^3*y^3*z + x^2*y^4*z - x*y^5*z - 2*y^6*z + 3*x^5*z^2 - x^4*y*z^2 - 3*x^3*y^2*z^2 - x^2*y^3*z^2 - 2*x*y^4*z^2 + 2*x^4*z^3 + 3*x^3*y*z^3 - 3*y^4*z^3 + 2*x^3*z^4 - 3*x^2*y*z^4 - 2*x*y^2*z^4 + 3*y^3*z^4 + 3*x^2*z^5 - 2*x*y*z^5 + 3*x*z^6 + y*z^6 + 2*z^7 - 1/140*x^6 + x^5*y + x^4*y^2 + 2*x^3*y^3 - 3*x^2*y^4 + x*y^5 - 2*y^6 + x^5*z + x^4*y*z + 2*x^3*y^2*z + 2*x^2*y^3*z + 3*x*y^4*z - 2*y^5*z - x^4*z^2 + x*y^3*z^2 + 3*x^3*z^3 + 3*x^2*y*z^3 + 3*x*y^2*z^3 + y^3*z^3 - 3*x^2*z^4 + 3*x*y*z^4 - 2*x*z^5 - 3*y*z^5 - 3*z^6 - 1/210*x^5 - 3/4*x^4*y - 3*x^3*y^2 - 3*x^2*y^3 - 3*x*y^4 - 3*y^5 - x^3*y*z - x^2*y^2*z + 2*x*y^3*z + 2*y^4*z + x^2*y*z^2 + 2*x*y^2*z^2 + y^3*z^2 - x^2*z^3 + 2*y^2*z^3 - 2*x*z^4 - 2*y*z^4 + 3*z^5 + 3*x^2*y^2 - x*y^3 - 3*y^4 + 1/3*x^3*z + x*y^2*z + y^3*z - x^2*z^2 - x*y*z^2 - 3*y^2*z^2 + 2*x*z^3 - 3*y*z^3 - z^4 - x*y^2 + 3*x*y*z - 2*y^2*z + y*z^2 + 3*z^3', '3*x^8 + x^7*y + 3*x^5*y^3 + 3*x^4*y^4 + 3*x^3*y^5 + 2*x^2*y^6 - x*y^7 + 3*x^7*z - x^6*y*z - x^5*y^2*z + x^4*y^3*z - 3*x*y^6*z - y^7*z - 2*x^6*z^2 + x^5*y*z^2 + 2*x^4*y^2*z^2 - x^2*y^4*z^2 + 3*x*y^5*z^2 + 2*y^6*z^2 + x^5*z^3 + 2*x^4*y*z^3 + x^3*y^2*z^3 - 3*x*y^4*z^3 - 2*y^5*z^3 - x^4*z^4 + 2*x^3*y*z^4 - x^2*y^2*z^4 + x*y^3*z^4 + 3*y^4*z^4 + 3*x^3*z^5 - 3*x^2*y*z^5 + x*y^2*z^5 - 2*y^3*z^5 - x^2*z^6 + 2*x*y*z^6 + x*z^7 + 3*y*z^7 - 11/2520*x^7 - 3*x^6*y - 2*x^5*y^2 - 2*x^3*y^4 + 3*x*y^6 + 2*y^7 + 2*x^6*z + 3*x^5*y*z - 3*x^4*y^2*z - 2*x^3*y^3*z - x*y^5*z - 3*y^6*z + x^5*z^2 + x^4*y*z^2 + x^3*y^2*z^2 - x^2*y^3*z^2 + x*y^4*z^2 + 3*y^5*z^2 - 2*x^4*z^3 - x^3*y*z^3 + 2*x^2*y^2*z^3 - 2*x*y^3*z^3 + 2*y^4*z^3 + x^2*y*z^4 - 3*x*y^2*z^4 + y^3*z^4 - 3*x^2*z^5 - x*y*z^5 + 3*y^2*z^5 + 2*x*z^6 - 2*y*z^6 - 3*z^7 - 1/315*x^6 - 2*x^5*y + 3*x^4*y^2 + 3*x^3*y^3 - x*y^5 - y^6 + 3*x^5*z + 3*x^4*y*z + 3*x^3*y^2*z + x^2*y^3*z + 3*x*y^4*z + 3*y^5*z + 2*x^4*z^2 - x^3*y*z^2 - x^2*y^2*z^2 - x*y^3*z^2 + 2*y^4*z^2 - 2*x^3*z^3 - 2*x^2*y*z^3 - x*y^2*z^3 - 2*y^3*z^3 + 3*x^2*z^4 - x*y*z^4 + 2*x*z^5 + 3*y*z^5 - 3*z^6 - 1/2*x^4*y + 2*x^3*y^2 - x^2*y^3 - 2*x*y^4 - y^5 - x^4*z - x^3*y*z - 2*x*y^3*z - y^4*z - 3*x^2*y*z^2 + 3*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + 3*x*z^4 + y*z^4 - 3*z^5 - x^2*y^2 - 2*x*y^3 + y^4 + x^3*z + 2*x^2*y*z + x*y^2*z - 2*y^3*z + x^2*z^2 + 2*x*y*z^2 - y*z^3 + y^3 + 3*y^2*z + 3*y*z^2']
  ok   R4 N5, component W4 (G_130 = beta = gamma = 0, eps = 0): a point of the component (residual equations vanish)
      timing: member+triples 0.5 s, stabilization 0.2 s
  ok   R4 N5, component W4 (G_130 = beta = gamma = 0, eps = 0) anchor (A): fixed {'G_130': 0, 'P_501': 0, 'P_302': 0, 'P_311': 0, 'P_510': 0, 'P_320': 0}, solved []; member closed with H and the prescribed graded data: True; a triple of I of local length 37 (ds), stabilization (37, 12)
      anchor generators: {'P': 'X^7 + 2*X^4*Z - X^2*Y^2 - X^2*Y*Z + X^2*Z^2 + 2*X*Z^3 + 2*X*Y^2 - Y^3 + 2*X*Y*Z + 2*Y^2*Z + 2*Y*Z^2 + Z^3', 'G': 'X^4*Y - X^2*Y^2 + X^2*Y*Z + X^2*Z^2 + 2*X*Y^2 - 2*Y^3 - 2*X*Y*Z - Y*Z^2 + Z^3', 'H': 'X^3*Z - 2*X*Y^2 - 2*Y^3 + X*Y*Z - 2*Y*Z^2'}
      anchor triple: ['-x^8 - x^7*y - 2*x^6*y^2 + 3*x^5*y^3 + x^4*y^4 + x^2*y^6 - 2*x*y^7 + 3*y^8 - x^7*z - 2*x^6*y*z + 2*x^5*y^2*z - 2*x^3*y^4*z - 2*x^2*y^5*z + 3*x*y^6*z + 2*y^7*z - x^6*z^2 + x^5*y*z^2 - 2*x^4*y^2*z^2 - x^3*y^3*z^2 + 3*x^2*y^4*z^2 + 3*x*y^5*z^2 - y^6*z^2 + 3*x^5*z^3 + x^4*y*z^3 + x^3*y^2*z^3 + 3*x^2*y^3*z^3 + x*y^4*z^3 - 2*y^5*z^3 + 2*x^4*z^4 - x^3*y*z^4 + 3*x*y^3*z^4 - 2*y^4*z^4 + 3*x^3*z^5 + 3*x^2*y*z^5 + 3*x*y^2*z^5 + 3*y^3*z^5 - x^2*z^6 - y^2*z^6 - x*z^7 - z^8 + 1/126*x^7 + 2*x^6*y + 3*x^5*y^2 + x^4*y^3 - 2*x^3*y^4 + 2*x^2*y^5 + 3*x*y^6 - 3*y^7 - 3*x^6*z - 3*x^4*y^2*z - 3*x^3*y^3*z + 2*x^2*y^4*z + x*y^5*z - 3*y^6*z - 2*x^5*z^2 + x^4*y*z^2 - x^3*y^2*z^2 - x^2*y^3*z^2 - 3*x*y^4*z^2 - 3*y^5*z^2 + 2*x^4*z^3 + 3*x^3*y*z^3 - x^2*y^2*z^3 - x*y^3*z^3 + 3*x^3*z^4 - 2*x*y^2*z^4 - y^3*z^4 - 2*x^2*z^5 - 2*x*y*z^5 - 2*y^2*z^5 + 2*x*z^6 - 2*z^7 + 1/24*x^6 - 3*x^5*y - x^4*y^2 + 2*x^3*y^3 + 3*x^2*y^4 + 2*x*y^5 - 2*y^6 - 3*x^5*z - 3*x^4*y*z - x^3*y^2*z - 3*x^2*y^3*z + 3*x*y^4*z + 3*y^5*z - x^4*z^2 - 2*x^3*y*z^2 + x^2*y^2*z^2 + 2*x*y^3*z^2 + 3*x^3*z^3 + x^2*y*z^3 + 3*x^2*z^4 + 3*x*y*z^4 - 3*y^2*z^4 - 3*x*z^5 + 2*y*z^5 + 2/3*x^4*y - 2*x^3*y^2 - 3*x^2*y^3 + 3*x*y^4 + y^5 - x^4*z + 3*x^3*y*z - y^4*z - 2*x^3*z^2 - 2*x^2*y*z^2 - x*y^2*z^2 + 2*x^2*z^3 + 2*x*y*z^3 + 3*y^2*z^3 + x*z^4 - z^5 - 5/12*x^3*y + 2*x^2*y^2 + 3*x*y^3 - 23/6*x^3*z + 3*x^2*y*z + 3*x*y^2*z + 3*y^3*z - 3*x^2*z^2 + x*y*z^2 - y^2*z^2 + x*z^3 - 2*y*z^3 + 3*z^4 - x*y^2 - 2*y^3 + 3*x*y*z + 3*y^2*z + 2*y*z^2 - 2*z^3', '3*x^8 - 3*x^7*y - 3*x^6*y^2 + x^4*y^4 - 2*x^3*y^5 + 3*x^2*y^6 - 3*x*y^7 + y^8 - 3*x^6*y*z - 3*x^5*y^2*z - x^4*y^3*z + 3*x^3*y^4*z + x^2*y^5*z - 2*x*y^6*z + 3*y^7*z - 2*x^6*z^2 + 2*x^5*y*z^2 - 3*x^4*y^2*z^2 + x^3*y^3*z^2 + 3*x^2*y^4*z^2 - x*y^5*z^2 + 2*y^6*z^2 - x^5*z^3 - x^4*y*z^3 + 2*x^3*y^2*z^3 - 2*x^2*y^3*z^3 - 2*x*y^4*z^3 - 3*x^4*z^4 + 3*x^3*y*z^4 - 3*x^2*y^2*z^4 + 2*x*y^3*z^4 + 3*y^4*z^4 - x^3*z^5 + x^2*y*z^5 + x*y^2*z^5 - 2*y^3*z^5 - x^2*z^6 - 2*x*y*z^6 + 2*y^2*z^6 - x*z^7 - y*z^7 + z^8 + 13/504*x^7 + 2*x^6*y - x^5*y^2 + 2*x^4*y^3 - x^3*y^4 + x^2*y^5 + 3*y^7 - 3*x^5*y*z + 3*x^4*y^2*z - 3*x^3*y^3*z + 3*x^2*y^4*z - 2*x*y^5*z + y^6*z - 3*x^5*z^2 - 2*x^4*y*z^2 + 3*x^2*y^3*z^2 + 3*x*y^4*z^2 - y^5*z^2 + 3*x^2*y^2*z^3 - x*y^3*z^3 + 2*x^3*z^4 + 3*x*y^2*z^4 - x^2*z^5 - 3*x*y*z^5 + y^2*z^5 + 2*x*z^6 + 2*y*z^6 - z^7 - 2/63*x^6 - 3*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - x*y^5 - y^6 - 3*x^5*z - x^4*y*z + 2*x^3*y^2*z + x^2*y^3*z + 3*x*y^4*z + y^5*z + 3*x^4*z^2 - x^3*y*z^2 + 2*x^2*y^2*z^2 - x*y^3*z^2 + 2*x^2*y*z^3 + 3*x*y^2*z^3 + y^3*z^3 + x^2*z^4 + 3*x*y*z^4 - 2*y^2*z^4 - y*z^5 + 5/6*x^4*y + x^3*y^2 + 2*x*y^4 - 3*x^4*z - 2*x^3*y*z - x^2*y^2*z + 3*y^4*z + 3*x^3*z^2 + x^2*y*z^2 - 3*x*y^2*z^2 - y^3*z^2 + 3*y^2*z^3 + x*z^4 - 2*z^5 + 2*x^2*y^2 + 2*x*y^3 - 2*y^4 + 10/3*x^3*z + 2*x^2*y*z - x*y^2*z + 3*y^3*z - 3*x*y*z^2 - 2*y^2*z^2 + x*z^3 + 2*y*z^3 - 3*z^4 + y^3 + 3*y^2*z + 2*y*z^2', '-2*x^8 + x^7*y - x^5*y^3 + x^4*y^4 - 3*x^3*y^5 - 3*x^2*y^6 + 2*x*y^7 - y^8 - x^7*z - 2*x^6*y*z + 2*x^5*y^2*z + 3*x^4*y^3*z + x^3*y^4*z - 3*x^2*y^5*z + x*y^6*z + 3*y^7*z - 2*x^6*z^2 + 3*x^5*y*z^2 - 2*x^4*y^2*z^2 + 3*x^2*y^4*z^2 + 3*x*y^5*z^2 + 2*y^6*z^2 - x^5*z^3 - x^3*y^2*z^3 + 3*x^2*y^3*z^3 - 3*x*y^4*z^3 + 2*y^5*z^3 - x^4*z^4 - x^3*y*z^4 + 3*x^2*y^2*z^4 - 3*x*y^3*z^4 - 2*y^4*z^4 - 3*x^2*y*z^5 - 3*x*y^2*z^5 + 2*x^2*z^6 - 2*x*y*z^6 - 3*y^2*z^6 - 2*x*z^7 + 2*y*z^7 + 2*z^8 - 1/60*x^7 - 3*x^6*y - 2*x^5*y^2 - x^3*y^4 - x^2*y^5 - 3*x*y^6 + 2*y^7 + x^6*z + 2*x^5*y*z - 3*x^4*y^2*z + 2*x^2*y^4*z + 2*x*y^5*z - 2*y^6*z - 3*x^5*z^2 + 2*x^4*y*z^2 - x^3*y^2*z^2 - x^2*y^3*z^2 - x*y^4*z^2 + 2*y^5*z^2 - x^4*z^3 + 3*x^3*y*z^3 + 2*x^2*y^2*z^3 + x*y^3*z^3 + y^4*z^3 + 3*x^3*z^4 + 2*x^2*y*z^4 - 2*x*y^2*z^4 - y^3*z^4 - 2*x^2*z^5 + 3*x*y*z^5 + 3*y^2*z^5 + 2*x*z^6 - y*z^6 + 2*z^7 + 3/280*x^6 - 2*x^4*y^2 - 3*x^3*y^3 + 2*x^2*y^4 + 3*x*y^5 + 3*x^5*z - x^3*y^2*z - x*y^4*z - 2*x^4*z^2 - 3*x^3*y*z^2 + 3*x^2*y^2*z^2 + 3*x*y^3*z^2 + 3*y^4*z^2 + 2*x^3*z^3 + x^2*y*z^3 + x*y^2*z^3 + 3*x^2*z^4 - 2*x*y*z^4 + 3*x*z^5 - 2*y*z^5 + z^6 + 2/3*x^4*y - x*y^4 - 2*y^5 + 3*x^4*z - x^3*y*z + 2*x^2*y^2*z - x*y^3*z - 3*y^4*z + 2*x^3*z^2 - 3*x^2*y*z^2 - 3*y^3*z^2 - 2*x*y*z^3 - 2*y*z^4 - 3*z^5 - 1/12*x^3*y + 3*x^2*y^2 - 3*x*y^3 - 7/6*x^3*z - 2*y^3*z - 3*x^2*z^2 - x*y*z^2 + 2*y^2*z^2 - 3*x*z^3 + 2*y*z^3 - 2*z^4 + x*y^2 + 3*x*y*z - 2*y^2*z - 2*y*z^2 + z^3']
  ok   R4 N5, component W5 (G_130 = beta = gamma = 0, linear): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.2 s
  ok   R4 N5, component W5 (G_130 = beta = gamma = 0, linear) anchor (A): fixed {'G_130': 0, 'P_501': 0, 'P_302': 0, 'P_311': 0}, solved ['G_211', 'G_103']; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 12)
      anchor generators: {'P': 'X^7 - 2*X^5*Y + 2*X^3*Y^2 - 4/21*X*Y^3 + X*Y^2 - 2*Y^3 + X*Y*Z + Y*Z^2 + 2*Z^3', 'G': 'X^4*Y - X*Y^2 - Y^2*Z - Y*Z^2 + 2*Z^3', 'H': 'X^3*Z - 2*X*Y^2 + 2*Y^3 + X*Y*Z + 2*Y^2*Z + Y*Z^2'}
      anchor triple: ['-2*x^8 - x^7*y - 3*x^6*y^2 + 3*x^5*y^3 + 2*x^4*y^4 - x^2*y^6 - x*y^7 - y^8 - 2*x^7*z + 3*x^5*y^2*z + 3*x^2*y^5*z - x*y^6*z - 2*x^6*z^2 - 3*x^5*y*z^2 + 2*x^4*y^2*z^2 - x^3*y^3*z^2 - x^2*y^4*z^2 + x*y^5*z^2 + 3*y^6*z^2 - 2*x^5*z^3 + 2*x^3*y^2*z^3 - 2*x^2*y^3*z^3 + 2*x*y^4*z^3 + 3*y^5*z^3 + x^4*z^4 + x^3*y*z^4 - x^2*y^2*z^4 + 3*x*y^3*z^4 + x^3*z^5 + 3*x*y^2*z^5 + 2*y^3*z^5 + 2*x^2*z^6 - x*y*z^6 - x*z^7 - 2*y*z^7 - z^8 - 437/5040*x^7 + x^6*y + 2*x^5*y^2 + 3*x^4*y^3 - x^3*y^4 + x^2*y^5 - x*y^6 - y^7 - 3*x^6*z + x^5*y*z - x^4*y^2*z + x^2*y^4*z + 2*x*y^5*z - y^6*z - 2*x^5*z^2 - x^4*y*z^2 + 2*x^3*y^2*z^2 + x^2*y^3*z^2 + 2*x*y^4*z^2 + y^5*z^2 - x^4*z^3 + 3*x^3*y*z^3 + 3*x^2*y^2*z^3 - 3*x*y^3*z^3 + 2*y^4*z^3 + 3*x^3*z^4 + 3*x^2*y*z^4 + 2*x*y^2*z^4 + y^3*z^4 - x^2*z^5 - 3*x*y*z^5 + y^2*z^5 - x*z^6 + 2*z^7 - 467/8820*x^6 - 2*x^5*y + 3*x^3*y^3 + 2*x*y^5 + 3*y^6 - 2*x^5*z + x^4*y*z - 2*x^3*y^2*z - 3*x^2*y^3*z + x*y^4*z + 2*y^5*z - x^3*y*z^2 + x^2*y^2*z^2 + 2*x*y^3*z^2 - y^4*z^2 - 3*x^3*z^3 - x^2*y*z^3 + 3*x*y^2*z^3 - y^3*z^3 - 3*x^2*z^4 - x*y*z^4 - y^2*z^4 - x*z^5 - y*z^5 - 3*z^6 - 1/210*x^5 - 17/12*x^4*y - 2*x^3*y^2 + x^2*y^3 + 2*x*y^4 - y^5 - 3*x^4*z + x*y^3*z - y^4*z - 2*x^3*z^2 - 3*x*y^2*z^2 + y^3*z^2 - x*y*z^3 + 3*x*z^4 + y*z^4 - z^5 - 3*x^2*y^2 - 3*y^4 + 13/6*x^3*z - 2*x^2*y*z - 3*y^3*z - 2*x^2*z^2 + 3*x*y*z^2 + y^2*z^2 - 3*x*z^3 + 3*z^4 + x*y^2 + y^3 - 3*x*y*z - 3*y^2*z - 3*y*z^2 + 2*z^3', '-3*x^8 - x^7*y + 3*x^6*y^2 + x^5*y^3 - x^4*y^4 + x^3*y^5 - x^2*y^6 - x*y^7 + 3*y^8 - x^7*z + 3*x^6*y*z - 3*x^5*y^2*z - x^4*y^3*z + 2*x^3*y^4*z - 3*x^2*y^5*z + 3*x*y^6*z + 3*y^7*z - 3*x^6*z^2 + 2*x^5*y*z^2 + 3*x^4*y^2*z^2 - x^3*y^3*z^2 + 2*x^2*y^4*z^2 + 3*x*y^5*z^2 - y^6*z^2 + 3*x^4*y*z^3 - x^3*y^2*z^3 + 2*x^2*y^3*z^3 + x*y^4*z^3 - y^5*z^3 + 2*x^3*y*z^4 + x^2*y^2*z^4 - 3*x*y^3*z^4 + 2*y^4*z^4 - 3*x^3*z^5 + 2*x^2*y*z^5 + x*y^2*z^5 + 2*y^3*z^5 - x^2*z^6 - x*y*z^6 - 3*y^2*z^6 - 2*x*z^7 - 3*y*z^7 - 3*z^8 - 59/630*x^7 - x^6*y - 3*x^5*y^2 + x^3*y^4 - x^2*y^5 - 2*x*y^6 + 3*y^7 + 3*x^6*z - 2*x^5*y*z - 2*x^4*y^2*z + 2*x^3*y^3*z - 3*x^2*y^4*z + x*y^5*z + y^6*z + 2*x^5*z^2 - 2*x^4*y*z^2 + 2*x^3*y^2*z^2 + x*y^4*z^2 + y^5*z^2 - x^4*z^3 - x^3*y*z^3 + 3*x^2*y^2*z^3 + y^4*z^3 + 3*x^3*z^4 + 3*x^2*y*z^4 - 2*x*y^2*z^4 - 2*y^3*z^4 + 2*x^2*z^5 + x*y*z^5 - y^2*z^5 + x*z^6 + y*z^6 - z^7 - 19/1260*x^6 - 2*x^5*y + 3*x^4*y^2 - 2*x^3*y^3 - x^2*y^4 + x*y^5 + x^5*z + x^4*y*z - x^3*y^2*z - 3*x^2*y^3*z + 3*x*y^4*z + y^5*z + 2*x^4*z^2 + x*y^3*z^2 - 3*x^3*z^3 + x^2*y*z^3 - x*y^2*z^3 - 2*y^3*z^3 + 2*x^2*z^4 + 3*x*y*z^4 + 3*y^2*z^4 + 2*x*z^5 + 2*y*z^5 - 2*z^6 + 1/70*x^5 - 5/12*x^4*y + 2*x^2*y^3 - x*y^4 + y^5 - x^4*z + 2*x^3*y*z - 2*x^2*y^2*z - 3*x*y^3*z + 3*y^4*z + 2*x^2*y*z^2 + 2*x*y^2*z^2 + y^3*z^2 - x*y*z^3 + 3*y^2*z^3 - x*z^4 + 3*y*z^4 + 3*z^5 - x^2*y^2 + 2*y^4 - 1/3*x^3*z + 2*x*y^2*z + 2*y^3*z - 2*x^2*z^2 + 2*x*y*z^2 + 2*y^2*z^2 + 3*x*z^3 + 2*y*z^3 - 3*x*y^2 - 2*x*y*z - 2*y^2*z', 'x^8 - x^7*y + x^6*y^2 - 3*x^5*y^3 + 3*x^4*y^4 - 2*x^3*y^5 - 3*x*y^7 + 3*x^7*z + x^6*y*z - x^5*y^2*z - x^4*y^3*z + 2*x^3*y^4*z + 3*x^2*y^5*z - x*y^6*z - 3*y^7*z + 3*x^6*z^2 - 2*x^5*y*z^2 + 2*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + 3*x^2*y^4*z^2 + 2*x*y^5*z^2 - 2*y^6*z^2 - x^5*z^3 + 3*x^4*y*z^3 - 2*x^3*y^2*z^3 - x*y^4*z^3 - 3*y^5*z^3 - 2*x^3*y*z^4 - 3*x*y^3*z^4 - 3*y^4*z^4 - 3*x^2*y*z^5 + 2*x*y^2*z^5 + y^3*z^5 + x^2*z^6 + x*y*z^6 - 2*y^2*z^6 - 3*y*z^7 - 2*z^8 - 449/3528*x^7 - 3*x^6*y - x^5*y^2 + 2*x^3*y^4 - 3*x^2*y^5 - 3*y^7 + 3*x^6*z - x^5*y*z + 3*x^4*y^2*z - 3*x^3*y^3*z - 2*x^2*y^4*z + x*y^5*z - y^6*z + x^5*z^2 + x^4*y*z^2 - 3*x^3*y^2*z^2 - x^2*y^3*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 - 3*x^3*y*z^3 - x*y^3*z^3 + 3*y^4*z^3 + x*y^2*z^4 + y^3*z^4 - 2*x^2*z^5 - 2*y^2*z^5 + x*z^6 + 3*y*z^6 - z^7 + 187/1764*x^6 - 3*x^5*y - 3*x^4*y^2 - 2*x^3*y^3 + 2*x^2*y^4 + 2*x*y^5 + y^6 + 2*x^5*z - 3*x^4*y*z - x^3*y^2*z - 3*x^2*y^3*z - 3*x*y^4*z - y^5*z - 2*x^4*z^2 + 3*x^2*y^2*z^2 + 3*x*y^3*z^2 + 2*y^4*z^2 + x^3*z^3 - x^2*y*z^3 - 3*x*y^2*z^3 - y^3*z^3 + 2*x^2*z^4 - y^2*z^4 - y*z^5 + z^6 - 1/210*x^5 + 23/12*x^4*y - x^3*y^2 - 2*x^2*y^3 + 2*x*y^4 - 2*y^5 - 2*x^4*z - 2*x^3*y*z - x^2*y^2*z - 3*x*y^3*z - y^4*z + x^3*z^2 + 2*x^2*y*z^2 + 2*x*y^2*z^2 - 3*y^3*z^2 + 2*x^2*z^3 + 3*x*y*z^3 + 3*y^2*z^3 + 2*x*z^4 - 2*y*z^4 + 3*z^5 - 3*x^2*y^2 - 3*x*y^3 - y^4 - 16/3*x^3*z + 3*x^2*y*z - 3*x*y^2*z + 3*y^3*z + 3*x^2*z^2 - x*y*z^2 + y^2*z^2 - x*z^3 + 2*z^4 + x*y^2 + 2*y^3 - 2*x*y*z + 3*y^2*z + y*z^2 - 3*z^3']
--- R4 N4
  ok   R4 N4: the graded normal form is closed with H = [1, 3, 6, 4, 3, 2, 1, 1]
  ok   R4 N4: one lift per pivot of D_4..D_7
    134 closedness conditions in 88 tail coefficients; 55 eliminated with constant pivots; residual equations:
       (-2) * G_220 * P_402
       (2/7) * P_402 * (2*P_510 - 7*G_211)
       (1/84) * (20*P_501^2*G_220 - 10*P_510*P_501*G_211 + 40*P_510*P_501*H_120 - 20*P_510^2*H_111 - 12*P_501*P_320 + 6*P_510*P_311 - 84*P_302*G_220 + 21*P_311*G_211 - 168*P_402*G_120 - 84*P_311*H_120 + 84*P_320*H_111)
       (1/63) * (30*P_510*P_501*H_030 - 10*P_510^2*H_021 - 126*P_402*G_030 - 63*P_311*H_030 + 42*P_320*H_021)
       (1/147) * (2*P_510^2*P_402 + 70*P_510*P_501*H_021 - 70*P_510^2*H_012 - 294*P_402*G_021 - 147*P_311*H_021 + 294*P_320*H_012)
       (1/84) * (8*P_510*P_402*G_202 - 30*P_510*P_501*G_103 + 40*P_510*P_501*H_012 - 120*P_510^2*H_003 + 63*P_311*G_103 - 168*P_402*G_012 - 84*P_311*H_012 + 504*P_320*H_003)
       (-4/7) * G_220 * P_402
       (4/49) * P_402 * (2*P_510 - 7*G_211)
  ok   R4 N4: on {P_402 != 0} the residual equations are triangular in ['G_220', 'G_211', 'G_120', 'G_030', 'G_021', 'G_012'] with diagonal (const)*P_402: a graph (irreducible)
  ok   R4 N4: on {P_402 = 0}: {10*P_510*P_501 - 21*P_311 != 0}: triangular in ['H_021', 'H_030', 'G_103', 'H_120']; {-10*P_510^2 + 42*P_320 != 0}: triangular in ['H_021', 'H_012', 'H_003', 'H_111']; then multiples of 5*P_501^2*G_220 - 21*P_302*G_220: ['(-1/21) * G_220 * (-5*P_501^2 + 21*P_302)']
  ok   R4 N4, component W1 (P_402 != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N4, component W1 (P_402 != 0) anchor (A): fixed {'P_402': 1}, solved ['G_220', 'G_211', 'G_120', 'G_030', 'G_021', 'G_012']; member closed with H and the prescribed graded data: True; a triple of I of local length 37 (ds), stabilization (37, 11)
      anchor generators: {'P': 'X^7 + X^4*Z^2 - 2*X*Y*Z^2 + 1/2*Z^4 + 2*X*Y^2 + Y^3 - X*Y*Z + Z^3', 'G': 'X^4*Z - 2*X*Y*Z + Z^3', 'H': 'X^3*Y - X*Y^2 - 2*Y^3 - 2*X*Y*Z - 2*Y^2*Z + 2*Y*Z^2 - Z^3'}
      anchor triple: ['-2*x^8 + 2*x^7*y - 3*x^5*y^3 - 2*x^4*y^4 - 2*x^3*y^5 - 3*x^2*y^6 - 3*x*y^7 + y^8 + 3*x^7*z - x^6*y*z + x^5*y^2*z + x^4*y^3*z - 2*x^3*y^4*z - 2*x^2*y^5*z + 3*x*y^6*z - y^7*z + x^6*z^2 - 3*x^5*y*z^2 + 2*x^4*y^2*z^2 + 3*x^3*y^3*z^2 + 2*x^2*y^4*z^2 - x*y^5*z^2 - 2*y^6*z^2 + x^5*z^3 - 2*x^4*y*z^3 - 2*x^3*y^2*z^3 + 3*x^2*y^3*z^3 + 3*x*y^4*z^3 + y^5*z^3 - 3*x^4*z^4 + 2*x^3*y*z^4 + x*y^3*z^4 - 3*y^4*z^4 - x^3*z^5 + 2*x^2*y*z^5 + x*y^2*z^5 + 2*y^3*z^5 + x*y*z^6 - 3*x*z^7 + 2*y*z^7 - 1/140*x^7 - x^6*y + 2*x^3*y^4 - 2*x^2*y^5 - 3*x*y^6 + 3*y^7 - 3*x^6*z + x^5*y*z - 2*x^4*y^2*z + 2*x^3*y^3*z + x^2*y^4*z + 3*x*y^5*z + y^6*z - 2*x^5*z^2 + x^4*y*z^2 + 3*x^3*y^2*z^2 - x^2*y^3*z^2 + 3*x*y^4*z^2 - 2*y^5*z^2 + x^4*z^3 - 2*x^2*y^2*z^3 + x*y^3*z^3 + y^4*z^3 - 3*x^3*z^4 - y^3*z^4 + 3*x^2*z^5 - 3*x*y*z^5 - y^2*z^5 - x*z^6 - z^7 + 1/420*x^6 - x^5*y - x^4*y^2 - 2*y^6 + 2*x^5*z + 2*x^4*y*z - x^3*y^2*z - x^2*y^3*z - x*y^4*z + x^4*z^2 - 3*x^3*y*z^2 + 2*x^2*y^2*z^2 - 2*x*y^3*z^2 - 3*x^3*z^3 + x^2*y*z^3 - 3*x*y^2*z^3 + 2*y^3*z^3 - x^2*z^4 + 3*x*y*z^4 - x*z^5 + 3*y*z^5 - 2*z^6 + 1/35*x^5 + 3*x^4*y - 3*x^3*y^2 - 2*x^2*y^3 + x*y^4 + 3*y^5 + 1/4*x^4*z + 3*x^3*y*z - 3*x^2*y^2*z - x*y^3*z + 3*y^4*z - 3*x*y^2*z^2 + 2*y^3*z^2 + 2*x^2*z^3 + x*y*z^3 + x*z^4 - 2*y*z^4 - z^5 + 10/3*x^3*y + 2*x^2*y^2 + 3*x*y^3 - 2*y^4 - 3*x^2*y*z + 3*x*y^2*z + 3*y^3*z - 3*x^2*z^2 + 3*x*y*z^2 - 3*y^2*z^2 - 3*x*z^3 - 3*y*z^3 - 2*z^4 + 3*x*y^2 + 3*y^3 - y^2*z + 3*y*z^2 - z^3', '-3*x^8 - 3*x^7*y + x^5*y^3 - x^4*y^4 + 2*x^3*y^5 - 3*x*y^7 + x^7*z - x^6*y*z - x^5*y^2*z + 2*x^4*y^3*z - 3*x^3*y^4*z + 2*x^2*y^5*z - 2*x*y^6*z - 3*y^7*z - 2*x^6*z^2 - x^5*y*z^2 + x^4*y^2*z^2 + 3*x^3*y^3*z^2 - 2*x^2*y^4*z^2 - 3*y^6*z^2 + x^4*y*z^3 + x^3*y^2*z^3 - 3*x^2*y^3*z^3 + 2*x*y^4*z^3 - 2*y^5*z^3 + x^4*z^4 + x^3*y*z^4 - 2*x*y^3*z^4 + 2*y^4*z^4 - x^3*z^5 - x^2*y*z^5 - 2*x*y^2*z^5 - y^3*z^5 - 2*x^2*z^6 - 3*y^2*z^6 + 3*x*z^7 + 2*z^8 + 17/1260*x^7 + 2*x^6*y + 3*x^5*y^2 + 3*x^4*y^3 + 2*x^3*y^4 + 2*x*y^6 + 3*y^7 + 2*x^6*z + 2*x^5*y*z - x^4*y^2*z + 3*x^3*y^3*z - x^2*y^4*z + x*y^5*z + y^6*z + x^5*z^2 + x^4*y*z^2 + 3*x^3*y^2*z^2 + 3*x^2*y^3*z^2 - x*y^4*z^2 - 3*y^5*z^2 - x^4*z^3 + 3*x^3*y*z^3 + 3*x^2*y^2*z^3 + 2*x*y^3*z^3 - 2*y^4*z^3 - 3*x^3*z^4 - 3*x*y^2*z^4 - 3*y^3*z^4 - x^2*z^5 - 2*y^2*z^5 + 2*x*z^6 + 2*y*z^6 - z^7 + 17/630*x^6 - x^5*y + 2*x^4*y^2 + 2*x^3*y^3 + 3*x^2*y^4 - 3*x*y^5 - x^5*z - x^4*y*z - 2*x^3*y^2*z + 3*x^2*y^3*z - 2*x*y^4*z + 2*y^5*z - 2*x^4*z^2 - x^3*y*z^2 + x*y^3*z^2 - 3*y^4*z^2 + x^3*z^3 + 2*x^2*y*z^3 + 3*x*y^2*z^3 - 2*y^3*z^3 + x^2*z^4 + x*y*z^4 - 3*y^2*z^4 + 3*x*z^5 + 2*y*z^5 + 3*z^6 + 3*x^4*y + 2*x^3*y^2 - x^2*y^3 - 3*x*y^4 - 3*y^5 + 11/12*x^4*z - x^3*y*z - 3*x^2*y^2*z - 3*x*y^3*z - 3*x^3*z^2 + x^2*y*z^2 + 3*x*y^2*z^2 + 3*y^3*z^2 + 2*x^2*z^3 + 2*x*y*z^3 + 2*y^2*z^3 + 2*x*z^4 + 3*y*z^4 - z^5 + 5/3*x^3*y + 3*x^2*y^2 - x*y^3 + y^4 + 2*x^2*y*z + 2*x*y^2*z + 3*y^3*z + 2*x*y*z^2 + y^2*z^2 + 2*x*z^3 - 2*y*z^3 + 3*z^4 + 2*x*y^2 + 2*y^3 + 2*x*y*z - 3*y^2*z - 2*y*z^2 - 3*z^3', '2*x^8 - 3*x^7*y + 2*x^6*y^2 - x^5*y^3 - 2*x^3*y^5 - 2*x^2*y^6 + 2*x*y^7 + y^8 - 2*x^7*z - 3*x^6*y*z + 3*x^5*y^2*z + 2*x^4*y^3*z + 2*x^3*y^4*z + 2*x*y^6*z + 2*y^7*z - x^6*z^2 + 3*x^5*y*z^2 + 2*x^4*y^2*z^2 + 3*x^3*y^3*z^2 + 3*x^2*y^4*z^2 - 3*x*y^5*z^2 + x^5*z^3 - 2*x^3*y^2*z^3 + 2*x^2*y^3*z^3 - 3*x*y^4*z^3 - 3*y^5*z^3 - 2*x^4*z^4 - 3*x^3*y*z^4 + 3*x^2*y^2*z^4 - x*y^3*z^4 + 2*y^4*z^4 - 2*x^3*z^5 + x^2*y*z^5 - 2*x*y^2*z^5 - 2*y^3*z^5 - x^2*z^6 - 3*y^2*z^6 - 3*x*z^7 + 2*y*z^7 - z^8 - 19/840*x^7 + 2*x^6*y + 3*x^5*y^2 + x^3*y^4 - 2*x^2*y^5 - 2*x*y^6 + y^7 - 2*x^6*z + 3*x^5*y*z + 2*x^4*y^2*z - 2*x^3*y^3*z + x^2*y^4*z - 3*x*y^5*z + 2*y^6*z + 3*x^5*z^2 - 2*x^4*y*z^2 + 3*x^3*y^2*z^2 - x^2*y^3*z^2 + x*y^4*z^2 + y^5*z^2 - 3*x^4*z^3 - 3*x^2*y^2*z^3 - x*y^3*z^3 + 3*y^4*z^3 - 2*x^3*z^4 + 3*x^2*y*z^4 - 2*y^3*z^4 - 2*x*y*z^5 - x*z^6 - 3*y*z^6 + z^7 + 11/1260*x^6 - 2*x^5*y - x^4*y^2 + 2*x^2*y^4 - 2*x*y^5 + 2*y^6 - 3*x^5*z + 2*x^4*y*z - 3*x^3*y^2*z - 3*x^2*y^3*z - 3*x*y^4*z - 2*y^5*z + 2*x^4*z^2 + 3*x^2*y^2*z^2 + 2*x*y^3*z^2 - 2*x^3*z^3 - x^2*y*z^3 + 3*x*y^2*z^3 - y^3*z^3 - x^2*z^4 - 2*x*y*z^4 - 2*y^2*z^4 - x*z^5 + 2*y*z^5 + 3*z^6 - 1/35*x^5 + x^4*y - 3*x^3*y^2 - 3*x^2*y^3 - 2*x*y^4 - y^5 + 1/2*x^4*z - 2*x^3*y*z - x^2*y^2*z - 3*x*y^3*z + 3*y^4*z - x^3*z^2 - 2*x^2*y*z^2 - x*y^2*z^2 - y^3*z^2 - x^2*z^3 + 2*x*y*z^3 - 2*y^2*z^3 + x*z^4 + y*z^4 + z^5 + 3*x^3*y + x^2*y^2 + 3*x*y^3 - y^4 + 3*x^2*y*z + 2*x*y^2*z + 3*y^3*z + 3*x^2*z^2 + 2*y^2*z^2 + 2*x*z^3 - y*z^3 + z^4 + 3*x*y^2 + y^3 + 2*y^2*z - y*z^2 - 2*z^3']
  ok   R4 N4, component W2 (P_402 = 0, beta' != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.2 s
  ok   R4 N4, component W2 (P_402 = 0, beta' != 0) anchor (A): fixed {'P_402': 0}, solved ['H_021', 'H_030', 'G_103', 'H_120']; member closed with H and the prescribed graded data: True; a triple of I of local length 39 (ds), stabilization (39, 12)
      anchor generators: {'P': 'X^7 + X^5*Z - 2*X^3*Y^2 - 2*X^3*Y*Z + 2*X^3*Z^2 + 20/7*X*Y^3 + 4*X*Y^2*Z + 13/7*X*Y*Z^2 - 2*X*Y^2 + 2*Y^3 + 2*X*Y*Z + Y^2*Z - Y*Z^2', 'G': 'X^4*Z - X*Y^2 + 2*Y^3 + X*Y*Z + 2*Y^2*Z + 2*Y*Z^2 + Z^3', 'H': 'X^3*Y - 15/7*X*Y^2 - 2*X*Y*Z'}
      anchor triple: ['-3*x^8 + 2*x^7*y + 3*x^6*y^2 + x^5*y^3 + 3*x^4*y^4 + 3*x^3*y^5 - 3*x^2*y^6 - x*y^7 + 3*y^8 - 3*x^7*z + 3*x^6*y*z - 2*x^4*y^3*z + 3*x^3*y^4*z + 2*x^2*y^5*z - 3*x*y^6*z + 2*y^7*z + 3*x^6*z^2 + 3*x^5*y*z^2 + 3*x^4*y^2*z^2 + x^2*y^4*z^2 - 2*x^5*z^3 - 2*x^4*y*z^3 + x^3*y^2*z^3 - 3*x^2*y^3*z^3 + y^5*z^3 + 2*x^4*z^4 + 2*x^2*y^2*z^4 - 2*x*y^3*z^4 + y^4*z^4 - x^3*z^5 - 2*x^2*y*z^5 + x*y^2*z^5 + x*y*z^6 + 3*y^2*z^6 + 3*x*z^7 - y*z^7 - 2*z^8 + 181/2205*x^7 - x^6*y - x^5*y^2 + x^3*y^4 + 3*x^2*y^5 - 2*x*y^6 - 3*y^7 - x^6*z + x^4*y^2*z + 2*x^2*y^4*z - 3*x*y^5*z - y^6*z - 2*x^5*z^2 - 3*x^4*y*z^2 + 3*x^3*y^2*z^2 - 2*x^2*y^3*z^2 - x*y^4*z^2 + y^5*z^2 + x^4*z^3 - 3*x^3*y*z^3 + x^2*y^2*z^3 - x*y^3*z^3 - 2*y^4*z^3 - 3*x^3*z^4 + 2*x^2*y*z^4 + 3*x*y^2*z^4 + 3*y^3*z^4 - x^2*z^5 + 3*x*y*z^5 - 2*y^2*z^5 + x*z^6 - y*z^6 + 2*z^7 + 173/35280*x^6 + 2*x^5*y - 2*x^4*y^2 + x^2*y^4 - 3*x*y^5 + 3*y^6 - 3*x^5*z - 3*x^4*y*z - x^3*y^2*z - x^2*y^3*z - 2*x*y^4*z - 3*y^5*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - 3*x*y^3*z^2 - 2*x^3*z^3 - x^2*y*z^3 + x*y^2*z^3 + y^3*z^3 - y^2*z^4 - 2*x*z^5 + y*z^5 - 1/420*x^5 + 2*x^4*y + x^3*y^2 - 3*x^2*y^3 + 3*x*y^4 - 2*y^5 - 13/24*x^4*z - 3*x^3*y*z - 2*x^2*y^2*z + x*y^3*z + 2*y^4*z - 3*x^3*z^2 - x^2*y*z^2 + 2*x*y^2*z^2 - 2*x^2*z^3 + 3*x*y*z^3 + y^2*z^3 - 2*x*z^4 - y*z^4 - z^5 - 2/7*x^3*y + 3*x^2*y^2 - 3*y^4 - 3*x^2*y*z + 2*x*y^2*z - y^3*z + 3*x^2*z^2 - x*y*z^2 + y^2*z^2 - x*z^3 + 2*y*z^3 - z^4 + x*y^2 - 3*x*y*z + y^2*z - y*z^2 + 3*z^3', '3*x^8 - 3*x^7*y + 3*x^6*y^2 - 2*x^5*y^3 + 3*x^4*y^4 + x^2*y^6 - 3*x*y^7 - 3*x^7*z + 3*x^6*y*z + 2*x^5*y^2*z - x^4*y^3*z - 3*x^3*y^4*z - 2*y^7*z - 3*x^6*z^2 - x^5*y*z^2 + 3*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + 3*x^2*y^4*z^2 + x*y^5*z^2 - y^6*z^2 + x^5*z^3 - 2*x^3*y^2*z^3 + 2*x^2*y^3*z^3 - x*y^4*z^3 + x^4*z^4 - 3*x^3*y*z^4 - 2*x^2*y^2*z^4 + x*y^3*z^4 - 2*y^4*z^4 + 3*x^3*z^5 - x^2*y*z^5 - 3*x*y^2*z^5 - y^3*z^5 + x^2*z^6 + 3*x*y*z^6 - y^2*z^6 + x*z^7 - 2*y*z^7 - z^8 - 263/17640*x^7 + x^6*y - 2*x^5*y^2 - 3*x^4*y^3 - 2*x^3*y^4 + x^2*y^5 + 2*x*y^6 - 3*y^7 - 2*x^6*z - 3*x^5*y*z + x^4*y^2*z - 2*x^3*y^3*z - x^2*y^4*z + 2*x*y^5*z + 3*y^6*z + 3*x^5*z^2 + 3*x^3*y^2*z^2 + 3*x*y^4*z^2 + 3*x^4*z^3 + x^3*y*z^3 - x^2*y^2*z^3 + x*y^3*z^3 + 3*y^4*z^3 - 3*x^2*y*z^4 - x*y^2*z^4 + y^3*z^4 + x^2*z^5 + 2*x*y*z^5 + 2*y^2*z^5 - 2*x*z^6 + 2*z^7 - 13/1680*x^6 - x^5*y + x^4*y^2 + x^3*y^3 + 3*x^2*y^4 + x*y^5 + 3*y^6 - x^4*y*z + 3*x^3*y^2*z + 2*x*y^4*z + y^5*z - 2*x^4*z^2 + 3*x^3*y*z^2 - 2*x^2*y^2*z^2 - 3*x*y^3*z^2 + 3*y^4*z^2 + 2*x^3*z^3 - x^2*y*z^3 + 3*x*y^2*z^3 - 2*y^3*z^3 + 2*x^2*z^4 - 3*x*y*z^4 - 3*y^2*z^4 - 3*x*z^5 - 3*y*z^5 - 2*z^6 + 1/84*x^5 - 3*x^3*y^2 + x^2*y^3 - 3*y^5 + 11/8*x^4*z + x^3*y*z + 2*x^2*y^2*z - 2*x*y^3*z + y^4*z - x^3*z^2 + 2*x*y^2*z^2 - 2*y^3*z^2 + x*y*z^3 - y^2*z^3 + 3*x*z^4 + 3*y*z^4 + z^5 + 37/21*x^3*y + 3*x*y^3 + 3*y^4 + 2*x^2*y*z + 2*x*y^2*z - 3*y^3*z - 2*x^2*z^2 + x*y*z^2 - 2*y^2*z^2 + 2*x*y^2 - 2*y^3 + x*y*z - 2*y^2*z - y*z^2 + z^3', '2*x^8 - x^7*y - x^6*y^2 - x^5*y^3 - 3*x^4*y^4 - 3*x^3*y^5 + 3*x^2*y^6 - 2*x*y^7 + 2*y^8 - 2*x^7*z + 3*x^6*y*z + 3*x^3*y^4*z + 2*x^2*y^5*z + 2*y^7*z + x^6*z^2 + 2*x^5*y*z^2 + x^4*y^2*z^2 - 3*x^2*y^4*z^2 + 3*x*y^5*z^2 - y^6*z^2 + 3*x^5*z^3 + 3*x^4*y*z^3 - x^3*y^2*z^3 + 2*x*y^4*z^3 - 2*y^5*z^3 + x^4*z^4 - x^3*y*z^4 + 3*x^2*y^2*z^4 - x*y^3*z^4 - y^4*z^4 - 3*x^3*z^5 + 2*x^2*y*z^5 + 3*x*y^2*z^5 + y^3*z^5 - 3*x^2*z^6 + 3*x*y*z^6 + 3*x*z^7 - 2*y*z^7 + 2*z^8 + 2/105*x^7 + 3*x^6*y - x^5*y^2 + x^4*y^3 + x^3*y^4 + 3*x^2*y^5 - 3*x*y^6 + y^7 + x^5*y*z + 3*x^3*y^3*z - 2*x^2*y^4*z - 3*x*y^5*z - 2*y^6*z - x^4*y*z^2 + 2*x^3*y^2*z^2 + 2*x^2*y^3*z^2 + y^5*z^2 + 3*x^4*z^3 - 2*x^3*y*z^3 - 2*x*y^3*z^3 + 2*y^4*z^3 - 2*x^2*y*z^4 + 2*y^3*z^4 + 3*x^2*z^5 - 2*x*y*z^5 - y^2*z^5 + 2*x*z^6 - 2*y*z^6 + 3*z^7 - 5/7056*x^6 - 2*x^5*y - 3*x^3*y^3 - 2*x^2*y^4 + 3*x*y^5 + 3*y^6 - x^4*y*z + 3*x^2*y^3*z + 2*y^5*z - x^4*z^2 - 2*x^3*y*z^2 - x^2*y^2*z^2 - 2*x*y^3*z^2 + 2*y^4*z^2 + x^3*z^3 - x^2*y*z^3 + 3*x*y^2*z^3 - 2*y^3*z^3 - 3*x^2*z^4 + 3*y^2*z^4 + 3*x*z^5 + 2*y*z^5 - 2*z^6 + 1/140*x^5 - 2*x^4*y + x^3*y^2 - 2*x^2*y^3 - 2*x*y^4 + 3*y^5 - 7/24*x^4*z - 2*x^3*y*z + x^2*y^2*z - 2*x*y^3*z - 2*y^4*z - x^3*z^2 + 2*x^2*y*z^2 + 2*y^3*z^2 - x^2*z^3 - x*y*z^3 - y^2*z^3 - x*z^4 - 2*y*z^4 + 23/21*x^3*y - 3*x^2*y^2 - 3*x*y^3 - 3*y^4 + x^2*y*z - x*y^2*z - 3*y^3*z - x^2*z^2 + 2*x*y*z^2 + y^2*z^2 - 2*x*z^3 + 2*z^4 + 2*x*y^2 - y^3 - x*y*z + 2*y^2*z + y*z^2 + 2*z^3']
  ok   R4 N4, component W3 (P_402 = 0 = beta', gamma' != 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.1 s
  ok   R4 N4, component W3 (P_402 = 0 = beta', gamma' != 0) anchor (A): fixed {'P_402': 0, 'P_510': 0, 'P_311': 0}, solved ['H_021', 'H_012', 'H_003', 'H_111']; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 11)
      anchor generators: {'P': 'X^7 + 2*X^5*Z + 2*X^3*Y^2 + 8/3*X*Y^3 - 2*Y^4 + 4/7*X*Y^2*Z - 2*X*Y^2 - 2*X*Y*Z + 2*Y^2*Z + 2*Y*Z^2', 'G': 'X^4*Z - X*Y^2 + Y^3 + 2*X*Y*Z + Z^3', 'H': 'X^3*Y + 2*X*Y^2 - 2*Y^3 + 2/7*X*Y*Z'}
      anchor triple: ['-3*x^8 - 2*x^6*y^2 + x^5*y^3 + x^4*y^4 + x^3*y^5 + 3*x*y^7 + 3*y^8 - 2*x^7*z - x^5*y^2*z + 3*x^4*y^3*z - 2*x^3*y^4*z - 3*x^2*y^5*z + 3*x*y^6*z + 2*y^7*z - 3*x^6*z^2 - x^5*y*z^2 + 3*x^4*y^2*z^2 - x^3*y^3*z^2 - 3*x^2*y^4*z^2 - 3*x*y^5*z^2 - 2*y^6*z^2 - 3*x^5*z^3 - 2*x^4*y*z^3 - 3*x^3*y^2*z^3 + 3*x^2*y^3*z^3 - y^5*z^3 + 3*x^4*z^4 - x^3*y*z^4 + 3*x^2*y^2*z^4 - 3*x*y^3*z^4 - 3*y^4*z^4 - 2*x^3*z^5 - 2*x^2*y*z^5 + 3*x*y^2*z^5 - 3*y^3*z^5 + 3*x^2*z^6 - x*y*z^6 + 3*y^2*z^6 - y*z^7 - 3*z^8 + 43/588*x^7 - 3*x^5*y^2 - 3*x^4*y^3 + x^2*y^5 - 2*x*y^6 - 3*y^7 + 2*x^6*z + x^5*y*z + 2*x^4*y^2*z - x^3*y^3*z - 2*x^2*y^4*z + x*y^5*z + 3*y^6*z + 3*x^5*z^2 + x^4*y*z^2 - 2*x^3*y^2*z^2 - 3*x^2*y^3*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 - 2*x^4*z^3 + x^2*y^2*z^3 + 3*x*y^3*z^3 + 2*y^4*z^3 + 2*x^3*z^4 - 2*x^2*y*z^4 - x*y^2*z^4 + x^2*z^5 + x*y*z^5 + 3*y^2*z^5 - 3*x*z^6 - 3*y*z^6 + z^7 + 8/735*x^6 - 2*x^5*y + x^4*y^2 + 2*x^2*y^4 + 3*x*y^5 + 2*y^6 - x^5*z - 2*x^2*y^3*z - y^5*z + 3*x^3*y*z^2 + x^2*y^2*z^2 + 3*x*y^3*z^2 - 2*y^4*z^2 + x^3*z^3 - 2*x^2*y*z^3 - 3*x*y^2*z^3 - 3*x^2*z^4 + x*y*z^4 - 3*y^2*z^4 + 3*x*z^5 + z^6 + 1/210*x^5 - 2*x^3*y^2 - 3*x^2*y^3 + 3*x*y^4 + 2*y^5 + 1/6*x^4*z - 2*x^3*y*z - 3*x^2*y^2*z + 3*x*y^3*z - 3*y^4*z - 3*x^3*z^2 - 3*y^3*z^2 - x^2*z^3 - 2*x*y*z^3 - y^2*z^3 - 3*x*z^4 - 2*y*z^4 - z^5 - 16/3*x^3*y - 2*x^2*y^2 - 3*x*y^3 + y^4 + 2*x^2*y*z + 3*x*y^2*z - 2*x^2*z^2 + 3*x*y*z^2 + 2*y^2*z^2 - 2*x*z^3 - 2*y*z^3 - z^4 - x*y^2 - 3*y^3 + y^2*z + y*z^2 + 2*z^3', '-2*x^8 - 3*x^7*y + x^6*y^2 + 2*x^5*y^3 - 2*x^4*y^4 - 3*x^3*y^5 - 2*x^2*y^6 - x*y^7 - y^8 - 2*x^6*y*z - 2*x^5*y^2*z + x^4*y^3*z - 2*x^3*y^4*z - 2*x^2*y^5*z + 3*x*y^6*z - y^7*z - x^6*z^2 - x^5*y*z^2 + x^4*y^2*z^2 + x^2*y^4*z^2 + x*y^5*z^2 - y^6*z^2 - 3*x^5*z^3 + 2*x^4*y*z^3 - 3*x^3*y^2*z^3 + x^2*y^3*z^3 - 3*x*y^4*z^3 + 3*y^5*z^3 + 2*x^4*z^4 + x^3*y*z^4 - x^2*y^2*z^4 - 2*x*y^3*z^4 - y^4*z^4 + 3*x^3*z^5 + x*y^2*z^5 - 3*x^2*z^6 - x*y*z^6 - 3*y^2*z^6 - x*z^7 + y*z^7 - 3*z^8 + 913/8820*x^7 + 3*x^6*y - x^5*y^2 + 2*x^4*y^3 - 2*x^2*y^5 - x*y^6 + 3*y^7 - 2*x^6*z - x^5*y*z - x^4*y^2*z - 2*x^3*y^3*z - y^6*z + 2*x^5*z^2 - 2*x^4*y*z^2 - 2*x^3*y^2*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 - x^4*z^3 + x^3*y*z^3 + x^2*y^2*z^3 - 2*x*y^3*z^3 - y^4*z^3 - 3*x*y^2*z^4 + 2*y^3*z^4 - 3*x^2*z^5 + 2*x*y*z^5 - 2*y*z^6 - 43/1764*x^6 + 3*x^5*y - 3*x^4*y^2 + 2*x^3*y^3 - x*y^5 - 2*x^5*z - 3*x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x*y^4*z - 3*y^5*z + x^4*z^2 - 2*x^2*y^2*z^2 + 3*x*y^3*z^2 - y^4*z^2 - x^3*z^3 + x*y^2*z^3 - y^3*z^3 - x^2*z^4 + x*y*z^4 + 2*y^2*z^4 - x*z^5 + 3*y*z^5 + 3*z^6 - 1/210*x^5 + x^4*y + 2*x^3*y^2 - 3*x^2*y^3 - x*y^4 - y^5 + 3/4*x^4*z + x^3*y*z - 3*x^2*y^2*z - 3*x*y^3*z - 3*y^4*z - 2*x^3*z^2 + 3*x^2*y*z^2 - 2*x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 - 3*x*y*z^3 - 2*y^2*z^3 + 2*x*z^4 + 2*y*z^4 - 3*z^5 - 32/7*x^3*y - x^2*y^2 + 2*y^4 + 2*x^2*y*z + 2*x*y^2*z + 2*y^3*z - 3*x^2*z^2 + 2*x*y*z^2 - 2*x*z^3 - 3*y*z^3 + z^4 + x*y^2 - 2*y^3 - 2*x*y*z - y^2*z + 2*y*z^2', '-x^7*y - 3*x^6*y^2 - 3*x^5*y^3 + 2*x^4*y^4 + 2*x^3*y^5 - 3*x^2*y^6 + 2*x*y^7 + 2*y^8 + 2*x^7*z + 3*x^6*y*z - 2*x^5*y^2*z + 2*x^4*y^3*z - 2*x^3*y^4*z + x^2*y^5*z + 3*x*y^6*z - 2*y^7*z + x^5*y*z^2 + 3*x^3*y^3*z^2 - 2*x^2*y^4*z^2 + x*y^5*z^2 - 3*y^6*z^2 + 3*x^4*y*z^3 - x^3*y^2*z^3 + 3*x^2*y^3*z^3 - 3*x*y^4*z^3 + y^5*z^3 + 3*x^4*z^4 - 3*x^3*y*z^4 - x^2*y^2*z^4 - 3*x*y^3*z^4 + 3*y^4*z^4 - 2*x^3*z^5 + x^2*y*z^5 + 3*x^2*z^6 + 2*x*y*z^6 + 2*y^2*z^6 - 2*x*z^7 + y*z^7 + 2*z^8 + 397/8820*x^7 + 3*x^6*y + x^5*y^2 - 2*x^2*y^5 + 2*x*y^6 + 2*y^7 - x^6*z - x^4*y^2*z - 2*x^3*y^3*z - 3*x^2*y^4*z + 2*x*y^5*z - y^6*z + 2*x^5*z^2 - 3*x^4*y*z^2 - x^3*y^2*z^2 - x^2*y^3*z^2 - 2*x*y^4*z^2 - 2*y^5*z^2 - 2*x^4*z^3 + 2*x^3*y*z^3 - 2*x^2*y^2*z^3 + y^4*z^3 - 3*x^3*z^4 + 3*x^2*y*z^4 - y^3*z^4 - 2*x^2*z^5 + 2*x*y*z^5 - 2*y^2*z^5 + 3*x*z^6 - 3*y*z^6 + z^7 + 37/1764*x^6 + x^5*y + x^4*y^2 + x^3*y^3 - 3*x^2*y^4 - 2*x*y^5 - 2*y^6 - x^5*z + 3*x^3*y^2*z - x^2*y^3*z + x*y^4*z - 2*y^5*z - 3*x^4*z^2 - 2*x^3*y*z^2 + x^2*y^2*z^2 + 2*x*y^3*z^2 + y^4*z^2 - 2*x^2*y*z^3 + 2*x*y^2*z^3 - 2*y^3*z^3 + 2*x*y*z^4 + 3*x*z^5 - 2*y*z^5 - 2*z^6 + 3*x^4*y + x^3*y^2 + x^2*y^3 + 2*y^5 - 3/4*x^4*z + x^3*y*z - 3*x^2*y^2*z + x*y^3*z - 3*y^4*z - 2*x^3*z^2 + 2*x^2*y*z^2 + 3*x*y^2*z^2 - 3*y^3*z^2 - 2*x^2*z^3 + y^2*z^3 - 3*x*z^4 - 2*y*z^4 - z^5 + 6*x^3*y + x^2*y^2 + 2*x*y^3 + y^4 + 3*x^2*y*z + x*y^2*z + x^2*z^2 - 2*x*y*z^2 + 2*y^2*z^2 - 2*z^4 + 3*y^3 + 2*y^2*z - y*z^2']
  ok   R4 N4, component W4 (P_402 = beta' = gamma' = 0, G_220 = 0): a point of the component (residual equations vanish)
      timing: member+triples 0.4 s, stabilization 0.2 s
  ok   R4 N4, component W4 (P_402 = beta' = gamma' = 0, G_220 = 0) anchor (A): fixed {'P_402': 0, 'P_510': 0, 'P_311': 0, 'P_320': 0, 'G_220': 0}, solved []; member closed with H and the prescribed graded data: True; a triple of I of local length 38 (ds), stabilization (38, 12)
      anchor generators: {'P': 'X^7 - 2*X^4*Y + X^2*Y*Z - X^2*Z^2 - 2*X*Z^3 - 2*X*Y^2 - 2*X*Y*Z + 2*Y^2*Z - 2*Y*Z^2 - 2*Z^3', 'G': 'X^4*Z - 2*X^2*Z^2 - X*Z^3 - X*Y^2 + Y^3 + 2*X*Y*Z - 2*Y*Z^2 + 2*Z^3', 'H': 'X^3*Y - 2*X*Y^2 + Y^3 + 2*X*Y*Z - Y*Z^2 + 2*Z^3'}
      anchor triple: ['-2*x^8 - x^7*y - 3*x^6*y^2 - 3*x^5*y^3 + 3*x^4*y^4 + 2*x^3*y^5 + 3*x*y^7 - 2*y^8 - x^7*z + x^6*y*z + 3*x^5*y^2*z - 3*x^4*y^3*z - 2*x^3*y^4*z + x^2*y^5*z - x*y^6*z - 2*y^7*z - x^5*y*z^2 - 3*x^4*y^2*z^2 + 2*x^3*y^3*z^2 + 2*x*y^5*z^2 - 2*y^6*z^2 - 3*x^5*z^3 + 3*x^4*y*z^3 + 2*x^3*y^2*z^3 + 3*x^2*y^3*z^3 - 2*x*y^4*z^3 + 2*y^5*z^3 + 3*x^4*z^4 - 2*x^3*y*z^4 - 3*x^2*y^2*z^4 + x*y^3*z^4 + 3*y^4*z^4 + x^2*y*z^5 - 2*y^3*z^5 - 3*x^2*z^6 - y^2*z^6 + 3*y*z^7 + 2*z^8 - 53/2520*x^7 + 3*x^6*y + x^5*y^2 - x^4*y^3 + 2*x^3*y^4 - 3*x^2*y^5 + 3*x*y^6 - 3*y^7 + 2*x^6*z - x^5*y*z + 3*x^4*y^2*z - 2*x^3*y^3*z - x^2*y^4*z - 2*x*y^5*z - 2*y^6*z + x^5*z^2 - 2*x^4*y*z^2 - 3*x*y^4*z^2 - 2*y^5*z^2 - x^4*z^3 + 3*x^3*y*z^3 - x*y^3*z^3 - 3*y^4*z^3 + 2*x^3*z^4 - x*y^2*z^4 - y^3*z^4 + 3*x^2*z^5 + 3*x*y*z^5 + y^2*z^5 + 3*x*z^6 + 3*y*z^6 - z^7 + 31/1260*x^6 - 3*x^5*y - x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - x^5*z + x^4*y*z + x^3*y^2*z + x*y^4*z + 2*y^5*z - 2*x^4*z^2 - 2*x^3*y*z^2 - 3*x^2*y^2*z^2 - x*y^3*z^2 + 3*y^4*z^2 - 3*x^3*z^3 + 3*x^2*y*z^3 + x*y^2*z^3 - 3*y^3*z^3 + x^2*z^4 + 2*x*y*z^4 - 3*y^2*z^4 - 2*x*z^5 + y*z^5 + 3*z^6 - 3*x^4*y + 2*x^3*y^2 + 2*x^2*y^3 - 2*x*y^4 + 3*y^5 + 23/12*x^4*z + 2*x^2*y^2*z - 2*x*y^3*z - 3*y^4*z + 3*x^2*y*z^2 + 2*y^3*z^2 + 2*x^2*z^3 - 3*x*y*z^3 - y^2*z^3 - 2*x*z^4 + y*z^4 - 2*z^5 + 8/3*x^3*y - 2*y^4 + 3*x^2*y*z + x*y^2*z + 2*y^3*z + 2*x^2*z^2 + x*y*z^2 + 3*y^2*z^2 + 3*x*z^3 + y*z^3 - z^4 + x*y^2 - 3*y^3 + 2*x*y*z + y^2*z - y*z^2', '2*x^8 - 2*x^7*y + 3*x^6*y^2 + 3*x^5*y^3 + 3*x^4*y^4 - 2*x^3*y^5 + x^2*y^6 + 3*y^8 + 3*x^7*z - 3*x^6*y*z + x^5*y^2*z - 3*x^2*y^5*z - y^7*z + 3*x^6*z^2 + 3*x^5*y*z^2 + 2*x^4*y^2*z^2 - x^3*y^3*z^2 + x^2*y^4*z^2 - x*y^5*z^2 - 2*y^6*z^2 - 3*x^5*z^3 + 3*x^2*y^3*z^3 - y^5*z^3 - 2*x^4*z^4 + 2*x^3*y*z^4 + x^2*y^2*z^4 + 3*y^4*z^4 + 3*x^3*z^5 + 2*x^2*y*z^5 - y^3*z^5 + 3*x^2*z^6 - x*y*z^6 + 2*y^2*z^6 - x*z^7 + 3*y*z^7 - 3*z^8 - 2/105*x^7 + 2*x^6*y - 2*x^5*y^2 - 2*x^3*y^4 + 3*y^7 + 3*x^6*z + x^5*y*z + 3*x^4*y^2*z + x^3*y^3*z + 2*x^2*y^4*z + 3*x*y^5*z + 3*y^6*z - 3*x^5*z^2 - x^4*y*z^2 + 3*x^3*y^2*z^2 - 3*x^2*y^3*z^2 - 3*x*y^4*z^2 + y^5*z^2 + 3*x^4*z^3 - 3*x^3*y*z^3 - 2*x^2*y^2*z^3 - 2*x*y^3*z^3 + x^3*z^4 - x^2*y*z^4 - 3*y^3*z^4 - 3*x^2*z^5 - x*y*z^5 - 2*y^2*z^5 + 3*x*z^6 - 2*y*z^6 - 3*z^7 - 13/840*x^6 + 2*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 + 3*x^2*y^4 - x*y^5 + 2*x^4*y*z - x^3*y^2*z + x^2*y^3*z + y^5*z - x^4*z^2 - 2*x^3*y*z^2 - 2*x^2*y^2*z^2 - 2*x*y^3*z^2 + y^4*z^2 - x^3*z^3 + x^2*y*z^3 + x*y^2*z^3 + 3*y^3*z^3 + x^2*z^4 + 2*y^2*z^4 + 2*y*z^5 - 3*z^6 - 2*x^4*y - 2*x^3*y^2 + 2*x^2*y^3 + x*y^4 - 3*y^5 - 1/3*x^4*z - 3*x^3*y*z - x^2*y^2*z - x*y^3*z - x*y^2*z^2 + 3*y^3*z^2 + 3*x^2*z^3 - 3*x*y*z^3 + y^2*z^3 - 3*x*z^4 - 2*y*z^4 - z^5 - 5/3*x^3*y + 2*x^2*y^2 + 2*y^4 + x^2*y*z + y^3*z + x^2*z^2 - 3*x*y*z^2 - 3*y^2*z^2 - 2*x*z^3 - 2*z^4 + 3*y^3 - x*y*z - 3*y^2*z + 3*y*z^2', '2*x^8 + 2*x^7*y + 2*x^6*y^2 + 2*x^5*y^3 - 2*x^4*y^4 - x^3*y^5 - x^2*y^6 - x*y^7 - 3*y^8 - 3*x^7*z - 3*x^6*y*z + 2*x^5*y^2*z - x^3*y^4*z + y^7*z - 2*x^6*z^2 + 2*x^4*y^2*z^2 - 3*x^3*y^3*z^2 - 3*x^2*y^4*z^2 + 3*x*y^5*z^2 + 2*y^6*z^2 - 3*x^5*z^3 - x^2*y^3*z^3 - 3*x*y^4*z^3 - 2*y^5*z^3 - x^4*z^4 + 2*x^3*y*z^4 + x^2*y^2*z^4 - 2*x*y^3*z^4 + 2*x^3*z^5 + 3*x^2*y*z^5 + x*y^2*z^5 + 3*y^3*z^5 - x^2*z^6 - 2*x*y*z^6 - y^2*z^6 + 2*x*z^7 + 3*y*z^7 - 2*z^8 - 23/2520*x^7 - 2*x^6*y - 3*x^5*y^2 - 3*x^4*y^3 - x^3*y^4 - x^2*y^5 - 3*x*y^6 - 2*x^5*y*z - 2*x^4*y^2*z + 2*x^3*y^3*z + x^2*y^4*z - 2*x*y^5*z + 3*y^6*z - 2*x^5*z^2 + x^4*y*z^2 - x^3*y^2*z^2 - 3*x^2*y^3*z^2 - 3*x*y^4*z^2 - 2*y^5*z^2 - x^3*y*z^3 + 2*x^2*y^2*z^3 + 3*x*y^3*z^3 - 2*y^4*z^3 + x^3*z^4 - x*y^2*z^4 + 2*y^3*z^4 + 2*x^2*z^5 - 3*x*y*z^5 - 2*x*z^6 - 2*z^7 - 13/2520*x^6 + 2*x^5*y - x^4*y^2 - 2*x^3*y^3 + 3*x^2*y^4 + 2*x*y^5 - 3*y^6 + x^5*z - 2*x^4*y*z - x^3*y^2*z + 3*x^2*y^3*z - 2*x*y^4*z - 3*y^5*z - x^3*y*z^2 - 3*x^2*y^2*z^2 - 3*x*y^3*z^2 - y^4*z^2 - 2*x^3*z^3 + x^2*y*z^3 + x*y^2*z^3 + x^2*z^4 - 3*x*y*z^4 - 3*y^2*z^4 + 2*x*z^5 + z^6 - 2*x^4*y - 3*x^3*y^2 - 2*x^2*y^3 + 2*x*y^4 - 2/3*x^4*z + x^3*y*z - 3*x^2*y^2*z - 3*y^4*z - x^3*z^2 + x^2*y*z^2 - 2*y^3*z^2 - x^2*z^3 - 3*x*y*z^3 + 2*y^2*z^3 - 2*x*z^4 - y*z^4 + 2*z^5 - x^3*y - 2*x^2*y^2 - 2*x*y^3 - 2*y^4 + 1/2*x^3*z + 2*x^2*y*z + 2*x*y^2*z - 3*y^3*z - 3*x^2*z^2 + y^2*z^2 + 3*x*z^3 - 3*y*z^3 + 2*x*y^2 - 2*y^3 + x*y*z + y^2*z + 2*z^3']
  ok   R4 N4, component W5 (P_402 = beta' = gamma' = 0, delta = 0): a point of the component (residual equations vanish)
      timing: member+triples 0.5 s, stabilization 0.2 s
  ok   R4 N4, component W5 (P_402 = beta' = gamma' = 0, delta = 0) anchor (A): fixed {'P_402': 0, 'P_510': 0, 'P_311': 0, 'P_320': 0, 'P_501': 0, 'P_302': 0}, solved []; member closed with H and the prescribed graded data: True; a triple of I of local length 37 (ds), stabilization (37, 12)
      anchor generators: {'P': 'X^7 + 2*X^4*Y - X^2*Y^2 - X^2*Y*Z + X^2*Z^2 + 2*X*Z^3 + Y^3 + X*Y*Z + 2*Z^3', 'G': 'X^4*Z + X^2*Y^2 + 2*X*Z^3 + Y^3 - 2*Y^2*Z + 2*Y*Z^2', 'H': 'X^3*Y + 2*X*Y^2 - Y^3 + 2*X*Y*Z + 2*Y^2*Z - 2*Z^3'}
      anchor triple: ['-3*x^8 - 3*x^7*y - 3*x^5*y^3 + x^3*y^5 - 3*x^2*y^6 + y^8 - 2*x^7*z + 3*x^6*y*z + x^4*y^3*z - x^3*y^4*z + 3*x^2*y^5*z + x*y^6*z - 3*y^7*z + x^5*y*z^2 - 2*x^4*y^2*z^2 + 3*x^3*y^3*z^2 + 3*x^2*y^4*z^2 - 2*x*y^5*z^2 + y^6*z^2 - x^5*z^3 - 3*x^3*y^2*z^3 + x^2*y^3*z^3 + 2*x*y^4*z^3 + 2*y^5*z^3 - x^4*z^4 - x^3*y*z^4 + x^2*y^2*z^4 + 3*x*y^3*z^4 - y^4*z^4 - 3*x^3*z^5 - x*y^2*z^5 + y^3*z^5 + 3*x^2*z^6 - x*y*z^6 + 2*y^2*z^6 + 3*y*z^7 - z^8 + 3/140*x^7 + 2*x^6*y + 2*x^4*y^3 + x^2*y^5 - 3*x*y^6 + 2*y^7 - 3*x^6*z + 2*x^5*y*z - 2*x^4*y^2*z - 3*x^3*y^3*z + x^2*y^4*z - 3*x*y^5*z + 3*y^6*z + 3*x^5*z^2 + x^4*y*z^2 - 3*x^3*y^2*z^2 - x^2*y^3*z^2 + 3*x*y^4*z^2 - 2*x^3*y*z^3 + 2*x^2*y^2*z^3 + 2*x*y^3*z^3 - 2*y^4*z^3 + 2*x^3*z^4 - x^2*y*z^4 + 2*y^3*z^4 - 3*x^2*z^5 + x*y*z^5 + 2*y^2*z^5 + 3*y*z^6 - 2*z^7 + 4/35*x^6 - 2*x^5*y - 2*x^4*y^2 + x^3*y^3 - 2*x^2*y^4 - 3*x*y^5 + y^6 + x^5*z - x^4*y*z - 3*x^3*y^2*z - 2*x^2*y^3*z - 3*y^5*z - x^4*z^2 - x^3*y*z^2 + 3*x^2*y^2*z^2 - 2*x*y^3*z^2 + 3*y^4*z^2 + x^3*z^3 - 2*x^2*y*z^3 + 3*x*y^2*z^3 - 3*y^3*z^3 + x^2*z^4 + 3*y^2*z^4 - x*z^5 - 2*y*z^5 - 2*z^6 - x^4*y + x^3*y^2 + 3*x^2*y^3 - x*y^4 + 3*y^5 + 13/12*x^4*z - 3*x^3*y*z + 2*x^2*y^2*z + x*y^3*z - 3*y^4*z + x^3*z^2 + 2*x^2*y*z^2 - y^3*z^2 + 3*x^2*z^3 + 3*x*y*z^3 + 2*y^2*z^3 - x*z^4 - 3*y*z^4 - 2*z^5 - 11*x^3*y - 2*x^2*y^2 + 2*x*y^3 + 3*y^4 + x^3*z - 3*x^2*y*z - 2*x*y^2*z - 2*x^2*z^2 - x*y*z^2 - 3*y^2*z^2 - x*z^3 + y*z^3 - 2*z^4 + 3*x*y^2 - 3*y^3 + 3*y*z^2 - 3*z^3', '2*x^8 - x^7*y - 2*x^6*y^2 - 3*x^5*y^3 + 2*x^4*y^4 - x^3*y^5 + x^2*y^6 - x*y^7 + 2*x^7*z + x^6*y*z + 3*x^5*y^2*z - x^4*y^3*z + 3*x^3*y^4*z - 2*x^2*y^5*z - x*y^6*z - y^7*z - 3*x^6*z^2 - x^5*y*z^2 - 2*x^4*y^2*z^2 + 3*x^3*y^3*z^2 - 3*x^2*y^4*z^2 + 2*x*y^5*z^2 - 2*y^6*z^2 + x^5*z^3 - 3*x^4*y*z^3 - 2*x*y^4*z^3 + 2*y^5*z^3 + 2*x^4*z^4 + 3*x^3*y*z^4 - 2*x^2*y^2*z^4 - 3*x^3*z^5 - 2*x^2*y*z^5 - 3*y^3*z^5 - 2*x^2*z^6 + 3*x*y*z^6 + y^2*z^6 - 2*x*z^7 + y*z^7 + z^8 - 1/126*x^7 - 3*x^6*y + 2*x^5*y^2 - x^4*y^3 - 3*x^3*y^4 - 2*x^2*y^5 + 3*x*y^6 - 2*y^7 - 3*x^6*z + 3*x^3*y^3*z + x^2*y^4*z - x*y^5*z + 3*y^6*z - 2*x^5*z^2 - x^3*y^2*z^2 - 2*x^2*y^3*z^2 + 3*x*y^4*z^2 - x^4*z^3 - 3*x^3*y*z^3 - x^2*y^2*z^3 + 2*x*y^3*z^3 - 3*x^3*z^4 + 2*x^2*y*z^4 - y^3*z^4 - 3*x^2*z^5 + 2*x*y*z^5 - 2*y^2*z^5 - y*z^6 - 2*z^7 - 13/1260*x^6 - 2*x^5*y - x^3*y^3 - 3*x^5*z - 2*x^4*y*z + x^3*y^2*z - 3*x^2*y^3*z + 3*x*y^4*z + 2*x^4*z^2 - x^3*y*z^2 + 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 - 2*x^2*z^4 - x*y*z^4 + 3*y^2*z^4 - 2*x*z^5 + y*z^5 - 3*z^6 + 3*x^3*y^2 + y^5 - 5/6*x^4*z + 3*x^3*y*z - 3*x^2*y^2*z - 3*x*y^3*z + y^4*z + 3*x^2*y*z^2 + 2*x*y^2*z^2 - 2*y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + z^5 + 2/3*x^3*y + x^2*y^2 + 2*x*y^3 + 2*y^4 - x^2*y*z + x*y^2*z + x^2*z^2 - x*y*z^2 - 2*y^2*z^2 + 3*x*z^3 - 2*z^4 - 3*x*y^2 - 2*y^3 + 2*x*y*z + y^2*z - y*z^2 + z^3', '3*x^8 - 2*x^5*y^3 - 3*x^4*y^4 + x^3*y^5 + 3*x^2*y^6 + 2*x*y^7 - 3*y^8 + 2*x^7*z - 2*x^6*y*z + 2*x^5*y^2*z + x^4*y^3*z + x^3*y^4*z + 2*x*y^6*z - 2*y^7*z + 3*x^6*z^2 - 2*x^5*y*z^2 - 2*x^4*y^2*z^2 - 2*x^3*y^3*z^2 - x*y^5*z^2 - 2*y^6*z^2 - x^5*z^3 - 2*x^4*y*z^3 + 3*x^3*y^2*z^3 - y^5*z^3 + x^4*z^4 - 2*x^3*y*z^4 + 2*x^2*y^2*z^4 - 3*x*y^3*z^4 + 2*y^4*z^4 - 3*x^3*z^5 - 2*x^2*y*z^5 + 3*y^3*z^5 - 2*x^2*z^6 - x*z^7 + 3*y*z^7 - 3*z^8 + 19/630*x^7 - 3*x^5*y^2 + 2*x^4*y^3 + 2*x^3*y^4 - x^2*y^5 + 2*x*y^6 - 3*y^7 - x^6*z + 2*x^5*y*z - x^4*y^2*z + x^3*y^3*z - 3*x^2*y^4*z + 3*y^6*z + 2*x^5*z^2 + 3*x^4*y*z^2 + 3*x^3*y^2*z^2 + x^2*y^3*z^2 - 2*x*y^4*z^2 - 3*y^5*z^2 + 3*x^4*z^3 - 3*x^3*y*z^3 - 2*x^2*y^2*z^3 + 3*x*y^3*z^3 + 3*y^4*z^3 + 3*x^3*z^4 - 3*x^2*y*z^4 + 3*x*y^2*z^4 + 2*y^3*z^4 - 2*x^2*z^5 + 3*y^2*z^5 + x*z^6 + 3*y*z^6 - 1/210*x^6 + x^4*y^2 - 2*x^3*y^3 - x^2*y^4 + 2*x*y^5 - y^6 - 2*x^5*z + 2*x^4*y*z - 3*x^3*y^2*z + 3*x^2*y^3*z - 3*x*y^4*z + 2*y^5*z + 3*x^4*z^2 + x^2*y^2*z^2 + 3*x*y^3*z^2 + y^4*z^2 + 3*x^3*z^3 - x^2*y*z^3 + x*y^2*z^3 - 2*y^3*z^3 + 3*x^2*z^4 + 2*y^2*z^4 + y*z^5 - z^6 - 3*x^4*y - 3*x^3*y^2 - x*y^4 + 2*y^5 + 3*x^4*z + 3*x^3*y*z - 2*x*y^3*z + 2*y^4*z + 2*x^3*z^2 - 3*x^2*y*z^2 + x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 - x*y*z^3 + 3*y^2*z^3 + x*z^4 + y*z^4 + z^5 - 2*x^2*y^2 - 2*x*y^3 - y^4 - x^3*z + 2*x^2*y*z - x*y^2*z - 3*y^3*z + 3*x^2*z^2 - 2*x*y*z^2 + y^2*z^2 - 3*x*z^3 + y*z^3 - 2*y^3 + 3*y^2*z - y*z^2 + 2*z^3']
time 21.0 s
anchor lengths: {'R4 N6': [39, 38], 'R4 N2': [36, 39, 39], 'R4 N3': [37, 39, 38, 37], 'R4 N7': [37, 39, 38, 37], 'R4 N1': [36, 36, 38, 38], 'R4 N5': [35, 36, 36, 37, 38], 'R4 N4': [37, 39, 38, 38, 37]}
ALL CHECKS PASSED
